NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS
|
|
- Jordan Martin
- 5 years ago
- Views:
Transcription
1 The Pennsylvania State University The Graduate School Department of Mathematics NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS A Dissertation in Mathematics by Thinh Duc Le c 2013 Thinh Duc Le Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013
2 ii The dissertation of Thinh Duc Le was reviewed and approved* by the following: Qiang Du Verne M. Willaman Professor of Mathematics Dissertation Adviser Chair of Committee Long-Qing Chen Distinguished Professor of Material Sciences and Engineering Ping Xu Distinguished Professor of Mathematics Mathieu Stienon Associate Professor of Mathematics Svetlana Katok Director of Graduate Studies, Department of Mathematics *Signatures are on file in the Graduate School.
3 iii Abstract Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries. Variations of these concepts have also been made over the years such as the discrete exterior calculus and the finite element exterior calculus. In this work, motivated by the recent studies of nonlocal vector calculus we develop a nonlocal exterior calculus framework on Riemannian manifolds which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal interactions (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Just as in the standard context, we introduce all necessary elements of exterior calculus such as forms, vector fields, exterior derivatives, etc. We point out the relationships between these elements with the known ones in (local) exterior calculus, discrete exterior calculus, etc. We also introduce nonlocal Hodge theory and its connections with existing works.
4 iv Table of Contents Acknowledgments vi Chapter 1. Introduction Local Exterior Calculus and Finite Element Exterior Calculus Discrete Exterior Calculus Existing Works in Nonlocal Vector Calculus Hodge Theory on Metric Spaces Details of this Dissertation Chapter 2. Nonlocal forms Oriented Simplices and Tuples Nonlocal Forms Nonlocal Exterior Derivative (D) Codifferential Operator (D ) NL Laplace-Beltrami Operator ( ) NL Hodge Operator ( ) Nonlocal Wedge Product ) nl 24 Chapter 3. Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus Nonlocal Trivializations (λ) Vector Fields NL Sharp ( ) and Flat ( ) Operators NL Vector Calculus
5 3.5 Some special cases of NL vector calculus operators v Chapter 4. Relationships to Local Geometry, Classical Vector Calculus and Discrete Vector Calculus Relationship between NL geometry and local geometry Relationship between NL Vector Calculus and the Local One Relationship between NL Laplacian and discrete Laplacian.. 46 Chapter 5. NL Hodge Theory Hodge Theory for the Discrete Derivative Hodge Theory for the Nonlocal Exterior Derivative Chapter 6. Another Model for Nonlocal Exterior Calculus in R n Nonlocal Forms Nonlocal Exterior Derivative (D) Codifferential Operator (D ) NL Laplace-Beltrami Operator (revisit) Hodge Operator ( ) Sharp ( ) and Flat ( ) Operators Vector Calculus (revisit) NL Wedge Product (Λ nl ) Chapter 7. Ongoing and Future Works Ongoing Works Future Works References
6 vi Acknowledgments I would like to thank Professor Ping Xu and Professor Aissa Wade for giving me the chance to study at Penn State. Without their great support I would not be able to come to Penn State for my Ph.D. I would like to thank my advisor, Professor Qiang Du, for believing and trusting in my capability by giving me the opportunity to be one of his students and investing his research projects and grants in me. He has been always available and willing to help during my studies. I really appreciate that he is very patient with me and gives me great supports on both research and finance. Without his brilliant guidance this dissertation could not be done. I would like to thank Professor Ping Xu and Professor Mathieu Stienon from the Department of Mathematics and Professor Long-Qing Chen from the Department of Material Sciences and Engineering for helpful discussions and serving on my committee. I also would like to thank Doctor Tadele Mengesha for his help via a lot of discussions with him in my final year at Penn State. I would like to give my appreciations to all the members in my research group for their helps and supports. Finally I would like to thank my family for their great supports especially when I am very far away from my home country.
7 1 Chapter 1 Introduction Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries [20]. They have made profound influence to the development of mathematics and they now can be learned from standard text books [33]. Variations of these concepts have also been made over the years. Highly successful examples include the discrete exterior calculus [10, 17] and the finite element exterior calculus [1] which are extensions to discrete spaces including piecewise linear complexes and finite element functions. They have proved to be useful in the development and analysis of finite element methods. More recently, motivated by the study of fractional diffusion processes and nonlocal electromagnetic media, the fractional exterior calculus has also been developed [9, 21, 34]. In this work, motivated by the recent studies of nonlocal calculus [16, 11], we develop a nonlocal (NL) exterior calculus framework which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal interactions (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Since our work is partly related to local exterior calculus, discrete exterior calculus, existing works on nonlocal vector calculus and a L 2 Hodge theory proposed by S. Smale et al. First of all we would like to give an literature overview on these theories which are relevant to our work.
8 2 1.1 Local Exterior Calculus and Finite Element Exterior Calculus The exterior calculus of differential forms, also called Cartan s calculus, is one whose geometric underpinning is the exterior algebra. It was born through a paper in 1899 by Élie Cartan ([6]). The first key ingredient of exterior calculus is differential forms. They are an approach to multivariable calculus that is independent of coordinates. Forms are local in the sense that they are defined pointwise: at any point x in a given manifold M, a p-form ω defines a skew-symmetric p-linear map ω : (T M) p R x x and as an operator of x, ω is required to be smooth (here T M is the tangent space x to M at x). Forms can interact with each other via wedge product (exterior algebra). Differential forms provide a unified approach to defining integrals over curves, surfaces, volumes and higher dimensional (Riemannian) manifolds. Another key ingredient is vector fields. These are the dual of 1-forms via musical isomorphisms (sharp and flat operators). Besides these two operators, other key operators include exterior derivative, codifferential, Hodge star, Laplace- Beltrami, etc. We call intrinsic properties the relationships between key ingredients and key operators which are coordinate-free. Following the same procedure, we define our nonlocal exterior calculus with similar key ingredients and key operators. Our goal is to preserve as many intrinsic properties as possible. However due to nonlocality (almost all operators are integral operators), some intrinsic properties may not be achievable and some operators play less important roles (see Section 1.5 below).
9 Exterior calculus has many applications, especially in geometry, topology, 3 partial differential equations (PDEs) and physics ([2],[13], [32]). Many partial differential equations (PDEs) are related to differential complexes, that is they can be rewritten using exterior calculus in neat forms. For example Maxwell s equations can be written very compactly in geometrized units using forms, exterior derivative and Hodge star. From this point of view, the authors of [1, 2] develop the theory of Finite Element Exterior Calculus, which serves the study on numerical analysis and scientific computation of many PDEs. These PDEs are related to differential complexes but they are a fundamental component of problems arising in many mathematical models. This theory is developed to capture the key structures of the L 2 de Rham complex and Hodge theory at the discrete level and to relate the discrete and continuous structures, in order to obtain stable finite element discretizations. The authors also develop an abstract Hilbert space framework (Hilbert complex), which captures key elements of Hodge theory and can be used to explore the stability of finite element methods. In our work we introduce the L 2 nonlocal de Rham complex, which is a Hilbert complex. We use some results on Hilbert complexes in [2] to study this nonlocal de Rham complex. 1.2 Discrete Exterior Calculus The authors of [10, 17] develop a theory of discrete exterior calculus (DEC) motivated by potential applications in computational methods for field theories such as elasticity, fluids, and electromagnetism. This discrete theory parallels the continuous (local) one in the sense that similar key ingredients and key operators are constructed (all in discrete forms) while some intrinsic properties are preserved.
10 4 The authors derive explicit formulas for some discrete differential operators in specific cases which are identical to the existing formulas in the literature. These formulas are proved to converge (in some sense) to their local (smooth) counterparts. In our work we first build the NL exterior calculus by following similar procedures as in the local or discrete exterior calculus. We also point out some relationships between the nonlocal exterior calculus and the local and the discrete ones. These relationships are either known in existing works (for example the convergence of NL operators to their local counterparts proved in [11], the use of the NL Laplacian as a bridge between the discrete one and the local one in [5]) or new (we propose new versions of operators in NL vector calculus in Chapter 3, a discretization of the NL Laplacian in Chapter 4). So our nonlocal exterior calculus is somewhere between the local one and the discrete one. 1.3 Existing Works in Nonlocal Vector Calculus In [14] the authors defines some nonlocal operators including NL partial derivatives, NL gradient, NL divergence, etc. These basic operators are then used to define new types of flows and functionals for image and signal processing (and elsewhere). This framework can be viewed as an extension of spectral graph theory([7], [23]) and the diffusion geometry framework ([8], [24]) to functional analysis and PDE-like evolutions. However, the discussion in these papers is limited to scalar problems. In [11] the authors develop a framework for nonlocal vector calculus, including the definition of nonlocal divergence, gradient, and curl operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are
11 5 also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators. Another application discussed in this paper is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations. Notice that in this work the authors define two different types of operators: point operators and two-point operators, where the latter ones are defined as adjoints of the former ones. In our work NL vector calculus is induced naturally from the nonlocal exterior calculus we develop just as in the standard context for the local one. That is vector calculus operators are defined using intrinsic relationships among key ingredients and key operators in exterior calculus (see Section 1.5 below). Thus we do not distinguish point operators and two-point operators (just as in local exterior calculus, there are only one gradient, one divergence and one Laplacian). Besides obtaining some similar basic operators and relationships among them as in [11], we also propose some new forms of these operators in Chapter 3 (Section 3.5). In some of these versions, two different horizon parameters are included, one is on the exterior derivative and the other is on the weight functions (also considered as on L 2 spaces of forms). We see this as having a potential of reducing the singularity of kernels of NL operators. 1.4 Hodge Theory on Metric Spaces The authors of [3, 30] develop a Hodge theory on metric spaces with two basic operators: a coboundary operator which is similar to the coboundary operator
12 6 of Alexander-Spanier cohomology ([31]) and a boundary operator, which is the L 2 - adjoint of the co-boundary operator. With these two operators the Hodge operator is defined, which is basically the Laplace-Beltrami operator. The authors develop an L 2 -Hodge theory and a Hodge theory at scale α. They also study α-harmonic forms, the α-cohomology and its relationship to the local de Rham cohomology. They also show that after rescaling, the α-laplacian they define converges to the smooth one under some assumptions as α goes to 0. Notice that for the Hodge theory at scale α, the authors use the coboundary operator of Alexander-Spanier cohomology and the complexes depend on α. This scale α plays the same role as the horizon parameter in NL peridynamics and NL vector calculus in [27, 29, 12, 35]. Now the discrete exterior derivative mentioned in Section 1.1 is also the coboundary operator of Alexander-Spanier cohomology. In our work, this is a special case of the nonlocal exterior derivative. In general the nonlocal exterior derivative is an integral operator. The codifferential is also defined as the adjoint of the exterior derivative with respect to a weighted L 2 inner product. The new approach in our work is that we embed the scale α (or the horizon parameter) into the weight functions and fix the complexes. Thus we only have one Hodge theory. Even though our work on Hodge theory is still in progress, it provides a different approach to the α-hodge theory. Also note that the local (smooth) exterior derivative is an unbounded operator. Our work can handle this kind of situation and we can still obtain the Hodge decomposition (and Poincaré inequality). Due to our definition of the NL exterior derivative we also provide different versions of the NL Laplacian which converge to the smooth (local) one without rescaling. Also from the approximation theory point of view as in [2], one might want to
13 7 keep the L 2 - de Rham complex as a whole (not depend on any parameter) and then find a finite element approximation of this complex (there is an approximation parameter involving). The scaled cohomology in [3, 30] already depends on the scale α, so if one would like to approximate it while keeping α fixed, there could be more difficulty. Our goal is to apply the approximation theory for a general Hilbert complex already developed in [2] for our NL de Rham complex to get similar results as in local exterior calculus. 1.5 Details of this Dissertation In chapter 2 we start with the discussion on nonlocal forms for oriented tuples defined on a Riemannian manifold. Nonlocal exterior derivative and codifferential operators are defined on forms, which are integral operators (in general). The L 2 spaces here are weighted. From these two operators we define the Laplace- Beltrami, Hodge star and wedge product. Due to nonlocality the last two operators play a less important role. In chapter 3 we first discuss about the nonlocal trivialization. The idea of this comes from [26]. Then nonlocal operators showing the duality between 1-forms and vector fields are defined. These are sharp and flat. Based on these operators some standard operators in vector calculus are defined including gradient, divergence and Laplacian. We show some relationships between local and nonlocal operators. Especially we show that the local (smooth) Laplacian can be approximated by different forms of the nonlocal Laplacian, which is an integral operator. Besides a well known form of the NL Laplacian which involve a single integral we offer a new form which involves a triple integral. In chapter 4 we show the relationships between nonlocal geometry and local geometry, nonlocal vector calculus and the local and discrete ones, especially
14 8 about Laplacian operators. We show some examples where the NL trivialization can induce the usual Levi-Civita connection. Moreover there are examples of NL trivializations which do not correspond to any Levi-Civita connection. We also offer a discrete version of the NL Laplacian which can give convergence under some assumptions. In chapter 5 we introduce our work (in progress) on the nonlocal Hodge theory, which uses the Hilbert complex theory presented in [2]. This NL Hodge theory mimics and partly extends the results about the Hodge theory presented in [3] (the L 2 Hodge theory results). This Hodge theory is believed to constitute a step towards understanding the geometry of vision ([3]). In chapter 6 we introduce a different model for nonlocal exterior calculus in the Euclidean space R n, which has some advantages over the model presented in previous chapters (for example it preserves more intrinsic properties and uses a natural inner product). We redefine all key objects and operators using the natural L 2 inner product. We cover some operators in vector calculus which already appear in previous chapters. Finally in chapter 7 we discuss about our ongoing and future works based on the work done in [2] for a general Hilbert complex. Recently NL calculus has been applied in studying image/signal processing and peridynamics. In [14] some NL operators (gradient, divergence, etc) are used to define some NL functionals. These functionals are called regularizing functionals, which replace the local notion of smoothness by the global notion of regularity. Based on these functionals some NL image/signal processing models are proposed. Due to the global regularity these models have been shown to have some advantage over similar local models such as the ability to detect and remove irregularities
15 9 from textures. In NL peridynamics theory ([27, 28, 29, 35]), there is no assumption on the differentiability of the displacement field, thus it has frequently been applied in the study of material failure. By studying NL operators systematically our framework can be used in these applications and for other purposes such as physical modeling of processes with NL behavior. The current notion of nonlocal exterior calculus also has many potential and natural applications. For instance, it can be used to study the approximations to classical (local) differential operators.
16 10 Chapter 2 Nonlocal forms In this chapter we define nonlocal forms and basic operators acting on forms. These operators have same names and meaning as those in standard (local) exterior calculus such as differential operator, codifferential operator, Hodge operator. Our goal is to define these operators in a way such that they preserve as many intrinsic properties as possible. To begin our discussion, let M be a Riemannian manifold of dimension n with a metric tensor g. For each point x M we denote the tangent space at x by T and x the inner product at x by g. We also fix a volume form on M (with respect to x the metric tensor). From now on all integrals will be with respect to this volume form. In fact in this chapter we only need this volume form (so M here can be a metric space). On the next chapter we will make use of the (local) inner product g. x 2.1 Oriented Simplices and Tuples Let e = (0,..., 0) be the origin and e (for i = 1,...,n) be the i th unit vector 0 i (consider as a point) in R n. For a nonnegative integer p the standard p-simplex p in R n is the convex hull of {e,..., e }. These points are vertices of the simplex. 0 p Two ordering of the vertices are equivalent if they differ from one another by an even permutation. If p > 0 then the orderings fall into two equivalence classes, each class is an orientation of the simplex. We simply write p = [e,..., e ] for 0 p the oriented simplex p with the equivalence class of the ordering (e,..., e ). 0 p
17 11 A singular p-simplex in the manifold M is a map s : p M (not necessary a one-to-one map). The points v = s(e ), i = 0,...,p are the vertices of the simplex in i i M. In order to define nonlocal exterior calculus we will actually only need the set of vertices (a tuple) of any simplex s in M and not the whole map s : p M. So we simply write s = [v,..., v ] for an oriented p-tuple s in M. Basically a p-tuple 0 p is just an element of the set M p+1. Definition We denote the real vector space generated by all oriented p- tuples in M by C p (M). Each element of this space is called a p-chain and is expressed as a finite formal sum of some oriented p-tuples (with coefficients in R). Remark If [v 0,..., v p ] is an oriented p-tuple then for any permutation σ in the symmetric group S p+1 we have [v σ(0),..., v σ(p) ] = sgn(σ)[v 0,..., v p ] (2.1) here sgn(σ) is the sign of the permutation σ. 2.2 Nonlocal Forms Definition A NL p-form is a map M p+1 R, which is skew-symmetric. That is ω(v σ(0),..., v σ(p) ) = sgn(σ) ω(v 0,..., v p ) (2.2) for any permutation σ S p+1. The vector space of all NL p-forms on M is denoted by Ω p NL (M). Remark A p-form can be extended linearly (over R) to a linear map C p (M) R. So skew-symmetry property means that if the orientation of a p- tuple is changed then the value of a p-form on this p-tuple also switches to the
18 12 opposite sign. We refer to [17] (p ) for more discussion about forms. 2. A 0-form is just a function f : M R. 3. In order to match up with the local exterior calculus, one can impose that all forms of order bigger than the dimension n is 0, that is Ω p (M) = {0} if p > n. NL 2.3 Nonlocal Exterior Derivative (D) Consider an oriented p-tuple [v 0,..., v p ] with p > 0. This tuple has p+1 faces which are oriented (p-1)-tuples where face i is [v 0,..., ˆv i,..., v p ]. Here the hat means we omit that vertex. Definition The discrete exterior derivative d : Ω p (M) Ωp+1 (M) p NL NL is a linear operator defined by p+1 (d ω)(v,..., v ) = ( 1) i ω(v,..., ˆv,..., v ) (2.3) p 0 p+1 0 i p+1 i=0 for any p-form ω and any (p+1)-tuple [v 0,..., v p+1 ]. Remark The discrete exterior derivative is just the couboundary operator of Alexander-Spanier cohomology ([31]). We refer to [17] for discussion about the discrete derivative in discrete exterior calculus. Notice that d 2 = 0, that is d d = 0 (we often omit the index p if there is no confusion). p+1 p Now in order to define the NL exterior derivative, for each index p we introduce a map T satisfying the following properties (for all p = 0, 1,...): p i) T p is a linear operator : Ω p NL (M) Ωp NL (M), that is T p preserves the skewsymmetry of forms. ii) T p+1 d p = d p T p (T is a chain map).
19 13 Let T = {T p : p = 0, 1,...} (T is a chain map by (ii) above). We see that the set of all T satisfying (i) and (ii) forms a real vector space. We denote this space by T (M). Also note that the identity map I : Ω p (M) NL p NL Ωp (M) satisfies NL (i) and (ii), thus I = {I : p = 0, 1,...} belongs to T (M). We can also intro- p NL duce the multiplication on T NL (M) as composition of two maps with the same index. Then the set of all invertible elements of T NL (M) forms a group (note that if T T NL (M) is invertible then T 1 T NL (M)). We see that T NL (M) is a unitary associative algebra over R. We usually write T instead of T p if there is no confusion. We now define the NL differential operator D. Definition Given T T (M). The nonlocal exterior derivative D : NL p Ω p (M) Ωp+1(M) is a linear operator defined as NL NL D p = d p T p That is for a form ω Ω p NL (M), (Dω)(v 0,..., v p+1 ) = p+1 i=0 ( 1)i (T ω)(v 0,..., ˆv i,..., v p+1 ) Proposition D 2 = 0, that is D p+1 D p = 0 Proof. By using property (ii) we see that D p+1 D p = d p+1 T p+1 d p T p = d p+1 d p (T p ) 2 = 0. Since T : Ω p (M) p NL Ωp (M) is a linear operator we know that (from the NL
20 14 Schwarz kernel theorem) its most general form is (T ω)(v,..., v ) = p 0 p M p+1 K p (v 0,..., v p ; v,..., 0 v ) p ω(v,..., 0 v ) p dv 0...dv, p where K : M 2p+2 R is some given kernel. Thus we have the following lemma: p Lemma T belongs to T NL (M) iff T preserves skew -symmetry of forms and the kernels K p, p = 0, 1,... satisfy the following identity p+1 ( 1) i K (v,..., ˆv,..., v ; v,..., p 0 i p+1 i=0 0 v ) = p p+1 ( 1) i K (v,..., v ; v,..., p+1 0 p+1 M i=0 0 v,..., p+1 v ) p dv, (2.4) p+1 for all (v,..., v ) and (v,..., 0 p+1 0 v ). Here on the i-th term of the sum on the right p hand side, v p+1 is at position i starting from the semicolon (;). Proof. For any ω Ω p (M), we have NL (T d ω)(v,..., v ) = p+1 p 0 p+1 = K (v,..., v ; v p+1,..., p+1 0 p+1 0 v ) ( 1) i ω(v,..., ˆv,..., p+1 i=0 0 i v ) p+1 dv 0...dv = p+1 = ( p+1 i=0 ( 1) i K (v,..., v ; v ),..., p+1 0 p+1 0 v,..., p+1 v ) ω(v,..., p 0 v ) p dv 0...dv p+1
21 15 (v p+1 is at position i starting from the semicolon (;)). Also p+1 (d T ω)(v,..., v ) = ( 1) i K (... ˆv...;... ˆv...) p p 0 p+1 p i i=0 i ω(v,..., ˆv,..., 0 i v ) p+1 dv... dv ˆ 0 i...dv p+1 = ( p+1 i=0 ( 1) i K (v,..., ˆv,..., v ; v ),..., p 0 i p+1 0 v ) ω(v,..., p 0 v ) p dv 0...dv p Comparing the results above we obtain the identity (2.4). We are still investigating the space T NL (M) to see if it has a canonical basis. In this manuscript, for computation and application purposes we usually use the following map T : Ω p (M) p NL Ωp (M) (and T = {T, p = 0, 1,...}) NL p ( p (T ω)(v,..., v ) = 0 p M p+1 K(v, v ) ) ω(v,..., i i=0 i 0 v ) p dv 0...dv, p ω Ωp NL (M). (2.5) Here K : M 2 R is a kernel function which is non-negative, symmetric and satisfies K(x, y) dy = 1 (2.6) M One can see that T preserves both symmetry and skew-symmetry of functions on M k+1. Also and we see that K (v,..., v ; v p+1,..., p+1 0 p+1 0 v ) = K(v, v ) p+1 i i=0 i K (v,..., v ; v,..., p+1 0 p+1 M 0 v ) p+1 dv = K i p (v 0,..., ˆv i,..., v p+1 ; v,..., ˆv,..., 0 i v ) p+1
22 16 (i = 0, 1,...). Thus T satisfies the identity (2.4) and it belongs to T NL (M). Remark We now list a few cases of interest of the operator D depending on the kernel K: 1. The kernel K is the Dirac delta function on M: K(x, y) = δ (y). In this case x the operator T is just the identity map and the NL differential operator is the discrete one (times a constant). 2. M is R n and the kernel K is a Gaussian kernel of the form K(x, y) = 1 (4πδ) n 2 e x y 2 4δ, here δ is a positive constant and x y is the Euclidean distance between x and y (notice that condition (2.6) holds and K is the heat kernel on R n ). In this case the operator D is truly an integral operator. For 0-forms (functions) on R n the operator T is the same as the well known Poisson transform (Gauss transform/gauss - Weierstrass transform is a special case when δ = 1). For a general Riemannian manifold M we can replace the kernel above by the heat kernel on M. The condition (2.6) means that the manifold M is required to be stochastically complete (see [15]). For example any compact Riemannian manifold is stochastically complete. 3. M is R n and the kernel K has the form C(δ) K(x, y) = x y s if x y δ 0 otherwise Again δ is a positive constant and C(δ) is the normalized constant such that condition (2.6) holds. The exponent s can be chosen in interval [0, n 1]. In this case D is also a truly integral operator.
23 Codifferential Operator (D ) In local exterior calculus the codifferential operator is the adjoint of the exterior derivative with respect to the standard inner product corresponding to the (local) volume form on the manifold M. Here first of all we assume that for each index p, Ω p NL (M) is equipped with an inner product (, ) p (we will specify one later). Definition i) Given T T NL (M), the (formal) adjoint of T is a linear operator T : Ω p NL (M) Ωp NL (M) defined by (T ω 1, ω 2 ) p = (ω 1, T ω 2 ) p, for any ω, ω Ω p (M) (p = 0, 1,...). 1 2 NL ii) The (formal) adjoint of the discrete derivative d is a linear operator d : p 1 Ω p (M) Ωp 1 (M) defined by (dω, η) NL NL p = (ω, d η), for any ω Ω p 1 (M), η p 1 NL Ω p (M) (p = 0, 1,...). NL iii) The codifferential operator D is the (formal) adjoint of the NL exterior derivative D. That is D p 1 : Ωp (M) Ωp 1(M) is a linear operator such NL NL that for any ω Ω p 1 NL (M), η Ωp NL (M). (Dω, η) p = (ω, D η) p 1 (2.7) Remark The codifferential operator can be computed by D = T d in some case (at least when d is bounded, see [36]). 2. We refer to [19] (p ) for the definition of the codifferential operator in local exterior calculus. 3. Because D 2 = 0 we also have (D ) 2 = 0.
24 We now introduce a weighted L 2 inner product on Ω p (M) for each index p. NL Weight functions: For each integer p = 0,...,n we fix a function W : M p+1 R p such that: i) W p (v 0,..., v p ) 0 for any (v 0,..., v p ) M p+1 18 ii) W p is symmetric, that is W p (v σ(0),..., v σ(p) ) = W p (v 0,..., v p ) for any permutation σ S. p+1 When p = 1 there are a few cases of interest for W. The trivial case is W The other cases are similar to examples 2 and 3 in Remark when M is a domain in R n : W has a Gaussian form 1 W 1 (x, y) = C(δ)e x y 2 δ, or C(δ) W (x, y) = x y s if x y δ 1 0 otherwise The constant C(δ) is chosen such that the following condition holds R n z 2 W 1 (z) dz = n (2.8) Here z = x y and W 1 is considered as a function of z. The exponent s can be chosen in interval [0, n + 1] We refer to [14] (p ) for more discussion about W 1 in R n. For a general manifold M one can use the heat kernel as in Remark or the Gaussian form K(x, y) = C(δ)e d(x,y)2 δ, where d(x, y) is the geodesic distance on M and C(δ) satisfies a condition similar to (2.8).
25 19 When p > 1 we usually define W p (v 0,..., v p ) = 0 i<j p W 1 (v i, v j ). From now on we will assume W 0 1. Definition For ω, η Ω p NL (M) we define the (W p -weighted) L2 -product of these two forms as (ω, η) = M p+1 ω(v 0,..., v p ) η(v 0,..., v p ) W p (v 0,..., v p ) dv 0...dv p (2.9) (again as mentioned at the beginning of this Chapter, all integrals are with respect to the fixed volume form given by the metric g). Remark If W p is positive everywhere (for example when W p has a Gaussian form) then (, ) is obviously an inner product in the L 2 sense. If this is not the case (for example when W is a cut-off function), in order to make (, ) an 1 inner product in the L 2 sense one can have two options: i) Redefine forms as following: for any form ω Ω p (M), ω(v NL 0,..., v p ) = 0 if W (v,..., v ) = 0 (this is well-defined because forms are skew-symmetric and p 0 p weight functions are symmetric). ii) Define an equivalence relation between any two p-forms ω, η as following: ω η iff ω(v 0,..., v p ) = η(v 0,..., v p ) whenever W p (v 0,..., v p ) 0. Then define Ω p (M) as the space of all equivalence classes of p-forms. NL 2. In [3] the weight function W is embedded into the definition of the exterior 1 derivative thus the authors only use the standard L 2 product (W p 1 for all p) We also define L 2 (Ω p (M)) as the space of all p-forms ω such that ω < NL, here. is the norm corresponding to the inner product above. This is a weighted L 2 space and so it is a Hilbert space (L 2 space of a measure on M)
26 Proposition The codifferential operator D corresponding to the operator T in (2.5) has an explicit formula as following: for any p-form ω, 20 (D p 1 ω)(v 0,..., v p 1 ) = ( 1) p (p + 1) W p 1 (v 0,..., v p 1 ) ( p 1 M p+1 K(v, v ) ) i i=0 i W p (v 0,..., v p ) ω(v 0,..., v p ) dv 0...dv p (2.10) (Of course D 0 by default). 1 In particular (when K is the Dirac delta function): (d ω)(v p 1 0,..., v p 1 ) = ( 1) p (p + 1) W (v,..., v ) ω(v,..., v ) dv. W (v,..., v ) p 0 p 0 p p p 1 0 p 1 M Proof. From the formula of D and the identity (2.7), by renaming variables inside integrals, one can obtain (D ω)(v 0,..., v p 1 ) = 1 W p 1 (v 0,..., v p 1 ) ( p ( 1) i K(v, v ) )...K(v 0 i=0 0 i, v )...K(v i+1 p 1, v ) W (v,..., p p 0 v ) p ω(v,..., 0 v ) p dv 0...dv p Here each product inside the sum always has p factors in order from v to v, 0 p 1 v 0 to v p and the variable v i is omitted. Since ω is skew-symmetric and W p is symmetric, all (p+1) terms inside the integral are equal and the term containing p 1 K(v i=0 i, v ) is associated with i ( 1)p. Hence we obtain the formula (2.10) above. Remark One can also define the operator D using the formula (2.10) then prove that it is formally adjoint to D.
27 NL Laplace-Beltrami Operator ( ) Definition As in standard context the NL Laplace-Beltrami operator : Ω p (M) Ω p (M) is defined as = DD + D D (= D p 1 D p 1 + D p D p ) 2.6 NL Hodge Operator ( ) As usual the Hodge operator is a linear operator : Ω p (M) Ωn p (M) NL NL (we assume that Ω p (M) = {0} for p > n). NL Here we define this operator as following: for any p-form ω, ( ω)(v 0,..., v n p ) = M p+1 K p (v 0,..., v n p, v n p+1,..., v n+1 ) ω(v n p+1,..., v n+1 ) dv n p+1...dv n+1 The kernel K p is a given function which is skew-symmetric with respect to (v 0,..., v n p ) and also skew-symmetric with respect to (v n p+1,..., v n+1 ). We would like to find condition(s) on the kernel functions such that the Hodge operator can maintain some intrinsic properties. One property we would like to have is D ω = ( 1) p D ω for any p-form ω.
28 Lemma Suppose that the differential operator D corresponds to T in (2.5). If the kernels K p (p = 0,..., n) are chosen such that the following condition 22 ( n+1 (p+2) M p+2 K(v, v ) ) K (v,..., v, v,..., i i=n p+1 i p+1 0 n p 1 n p v ) n+1 dv n p...dv n+1 = ( 1) n (n p+1) W n p 1 (v 0,...,v n p 1 ) M n p+1 ( n p 1 i=0 K(v, v ) ) W (v,..., i i n p 0 v ) K n p p (v,..., 0 v, v n p n p+1,..., v n+1 ) dv 0...dv n p is satisfied for all p = 0,..., n 1 and all (v 0,..., v n p 1, v n p+1,..., v n+1 ) then the identity D ω = ( 1) p D ω holds for any p-form ω. Proof. We will show direct calculation using the formulas of D and D in Sections 2.3 & 2.4. On one hand ( D ω)(v,..., v ) = 0 n p 1 n+1 ( 1) i (n p) K (v,..., v, v,..., v )(T ω)(v,..., ˆv,..., v ) p+1 0 n p 1 n p n+1 n p i n+1 i=n p dv n p...dv n+1 Since K p+1 is skew-symmetric with respect to the second component (v n p,..., v n+1 ), by renaming variables we can see that all terms in the sum above are equal and thus
29 23 ( D ω)(v 0,..., v n p 1 ) = (p + 2) ( n+1 K(v, v )K ) (v,..., v, v,..., v ) ω(v,..., i i=n p+1 i p+1 0 n p 1 n p n+1 n p+1 v ) n+1 dv n p...dv n+1 dv n p+1...dv n+1 On the other hand ( n p 1 i=0 (D ω)(v 0,..., v n p 1 ) = ( 1) n (n p + 1) W n p 1 (v 0,..., v n p 1 ) K(v i, v i ) ) W n p (v 0,..., v n p ) K p (v 0,..., v n p, v n p+1,..., v n+1 ) ω(v n p+1,..., v n+1 ) dv 0...dv n+1 Rename (v n p+1,..., v n+1 ) to (v n p+1,..., v n+1 ). Comparing two expressions above we see that in order for the identity D ω = ( 1) p D ω to hold for any p-form ω the condition in the statement of the lemma is sufficient. Remark In local exterior calculus, the Hodge operator (*) is an isomorphism at each point on M (between the p-th and (n-p)-th exterior products of the cotangent space at the given point, these vector spaces have finite dimensions). Moreover = ( 1) p(n p), thus the inverse of : Λ p (T M) x x Λn p (T M) is x ( 1) p(n p) x : Λ n p (T x M) Λp (T x M).
30 Here due to the nonlocality there is no guarantee that the NL Hodge still possesses these properties Nonlocal Wedge Product nl ) In order to define the wedge product of NL forms we can use the definition proposed by Castrillon-Lopez for discrete exterior calculus (see [10], p. 49 or [17], p ). Definition The wedge product of a p-form ω and a q-form η is a (p+q)-form given by (ω nl η)(v 0,..., v p+q ) = 1 (p + q + 1)! σ S p+q+1 sgn(σ) ω(v σ(0),..., v σ(p) ) η(v σ(p),..., v σ(p+q) ) We can verify that (ω η) is indeed a form (it is skew-symmetric) and satisfies nl the following properties: i) Anti-commutativity (ω η) = ( 1) pq (η ω) nl nl ii) Leibniz rule d (ω η) = (d ω) η + ( 1) p ω (dη) nl nl nl iii) Associativity for closed forms For a p-form ω, a q-form η and a r-form γ such that dω = 0, dη = 0, dγ = 0, we have (ω nl η) nl γ = ω nl (η nl γ) We refer to [10] (p and p. 49) for the proof. Remark The wedge product defined here only works for the discrete derivative d.
31 25 Chapter 3 Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus In this chapter we introduce NL trivializations on the manifold M which could be reduced to the Levi-Civita connection in some special case (see next chapter). We also define NL vector fields besides the usual vector fields on M. We then define NL operators showing the relationship between (NL) forms and vector fields just as in the standard context. With the help of these operators we can construct (NL) vector calculus. 3.1 Nonlocal Trivializations (λ) As mentioned in [26] a NL trivialization is a means to compare vector fields at any two points on manifold M directly, without any primary notion of infinitesimal transport of vectors or the accompanying path-dependent parallel transport. Definition A nonlocal trivialization λ is a map which corresponds any ordered pair of points (x, y) a linear isomorphism λ : T T such that: xy x y i) λ is the inverse of λ : λ = λ 1 yx xy yx xy ii) λ : T T is the identity map xx x x iii) λ xy preserves the inner products on T x and T y Remark We refer to [26] for more discussions about NL trivialization(s). The author in that manuscript only imposes condition (i) for a nonlocal trivialization.
32 26 2. There is no requirement about the continuity or smoothness of the NL trivialization. 3. Consider the case when the manifold M is the Euclidean space R n. The trivial NL trivialization on R n is just the identity map λ = Id : R n R n for any pair xy (x, y). An example of a nontrivial NL trivialization can be obtained as follows: for any pair (x, y) with x y, λ : R n R n is a given reflection (not necessary in a xy hyperplane). Another example: for any pair (x, y) with x y, λ : R n R n is the reflection xy in the hyperplane through the origin, orthogonal to the vector (x y). 4. Consider another example when M is the two-dimensional sphere S 2 in R 3. One can construct two NL trivializations on S 2 as follows: First of all for each x S 2, T is the plane through the origin O R 3 and perpendicular to x (so T = T ). If x = ± y then λ is the identity map Id : T T. x x xy x x x If x ± y there are two planes bisecting the angle Oxy and its supplementary. Define Ref to be the reflection in one of these planes and let λ be the restriction xy of Ref to T. One can verify that (x y) and (x + y) are the normal vectors to x the two planes above so λ xy : T x T y, λ xy (v) = v 2 < v x ± y > x ± y 2 (x ± y), v T x (3.1) (< > and. are the Euclidean inner product and norm in R 3 ). It is clear that λ xy is orthogonal and λ yx = λ 1 xy. From now on we assume that the manifold M is equipped with a NL trivialization λ. We call the geometry induced on M by a NL trivialization as NL
33 Riemannian geometry. Some of them will be discussed in subsequent sections. We refer to [26] for more details on NL differential geometry discussed there Vector Fields We refer to a usual vector field u : M x u(x) T x as a point vector field. We define a two-point (or NL) vector field V as a map V : M M (x, y) V (x, y) T x Definition A two-point vector field V is called i) λ-symmetric if λ (V (x, y)) = V (y, x) xy ii) λ-skew-symmetric if λ (V (x, y)) = V (y, x) xy here λ is a given NL trivialization on M as in Section 3.1. Since the NL trivialization λ is given from now on we will only write symmetric/skewsymmetric instead of λ-symmetric/λ-skew-symmetric. Remark Again there is no requirement about continuity or smoothness of vector fields here. 2. Two-point vector fields can be created from point-based vector fields by using NL trivialization λ. For example if u : x u(x) T is a point-based vector x field then V : (x, y) V(x,y) = λ (u(y)) T is a two-point vector field. Also yx x V (x, y) = u(x) λ (u(y)) is a two-point vector field which is skew-symmetric. yx 3.3 NL Sharp ( ) and Flat ( ) Operators As in the standard context, sharp and flat are operators showing relationships between 1-forms and NL (two-point) vector fields. First of all we start with
34 a general definition for each of these operators and then we will use a specific one for computation and application purposes. 28 Definition i) The sharp operator ( ) is a linear operator which maps any 1-form ω to a NL vector field ω defined by ω (x, y) = M 4 σ (x, y, x, y, x, y ) ω(x, y ) λ x (V (x, y )) dx dy dx dy. x 0 Here σ : M 6 R is a given kernel and V is a given NL vector field (λ is the 0 NL trivialization). ii) The flat operator ( ) is a linear operator which maps any two-point vector field V to a 1-form V defined by V (x, y) = M 4 σ (x, y, x, y, x, y ( ) g x V (x, y ), λ x x (V (x, y ) )) dx dy dx dy. 0 Here σ : M 6 R is a given kernel and V 0 is a given NL vector field. Remark Sharp and flat are linear operators between Ω 1 (M) and the NL (real) vector space generated by all NL vector fields on M. 2. Note that ω (x, y) T x thus in the definition (i) we need to use the NL trivialization so that λ x x (V 0 (x, y )) T x (similarly for the definition (ii)). For the relationship between sharp and flat we would like to preserve the following properties just as in standard context and thus put some constraints on the given data (σ, σ, V 0, V 0 ): 1) V has to be a 1-form (skew-symmetric function on M 2 ). 2) (ω ) = ω for any 1-form ω and (V ) = V for any NL vector field V. That is sharp and flat are the inverse of each other. However since 1-forms are scalar
35 29 functions and vector fields have dimension n, two identities above may not be achievable simultaneously. 3) g (ω ) (x, y), V (x, y) = ω(x, y)v (x, y) for all ω, V and (x, y). If this identity x happens we define this quantity as the pairing < ω, V > of ω and V at (x, y). Also < V, V 1 2 > (x, y) = g x (V 1 (x, y), V 2 (x, y)) for any two vector fields V 1, V 2. Note that due to nonlocality we may not be able to preserve all properties as in local exterior calculus. We now use a simple form of sharp and flat for computation and application purposes. First of all since 1-forms are skew-symmetric, their dual (NL vector fields) must have a similar property. Here we will define the dual of a 1-form as a symmetric NL vector field. To do that we fix a skew-symmetric vector field V : (x, y) V (x, y) T such that 0 0 ) x g (V (x, y), V (x, y) = 1 if x y (this means V (x, y) 0 if x y ). Here x we assume that such a vector field exists on M (note that there is no requirement about continuity or smoothness). For example if M is R n with the trivial NL trivialization one can take V 0 (x, y) = (y x)/ y x if x y (and V 0 (x, x) = 0). Another example: consider the sphere S 2. We write x S 2 as x = (x 1, x 2, x 3 ) and let (0, x, x ), 3 2 if x = 0 1 u (x) = 0 1 x 2 ( x, x, 0), x2 2 if x 0 1 Now we define
36 x y, V (x, y) = 0 u (x), 0 if x ± y if x = y 30 0, if x = y (Here is the cross product in R 3 ). One can check that V (x, y) T and with the NL trivialization(s) defined in 0 x Section 3.1 for S 2, λ (V (x, y)) = V (y, x) and V (x, y) 0 if x y. xy Definition i) The pairing between a 1-form ω and a two-point vector ) field V is defined as < ω, V > (x, y) = ω(x, y) g (V (x, y), V (x, y). x 0 ii) The sharp operator maps any 1-form ω to a symmetric two-point vector field ω defined by ω (x, y) = ω(x, y)v (x, y). 0 iii) The flat operator maps any symmetric two-point vector field V to a 1-form V defined by V ) (x, y) = g (V (x, y), V (x, y). x 0 Lemma i) The pairing < ω, V > is bilinear and non-degenerate with respect to ω. ii) ω is a symmetric vector field. iii) V is a 1-form. iv) < ω, V > (x, y) = g (ω ) (x, y), V (x, y) x = ω(x, y)v (x, y) (x, y), for any 1-form ω and any symmetric vector field V. v) (ω ) = ω, for any 1-form ω. Proof. i) It is clear that if < ω, V >= 0 for any V then ω = 0. ii) λ (ω ) ) ( )( (x, y) = ω(x, y) λ (V (x, y) = ω(y, x) xy xy 0 ) V (y, x) 0 = ω (y, x). iii) V ) ( ) (y, x) = g (V (y, x), V (y, x) = g λ (V (x, y)), λ (V (x, y)) y ) 0 y xy xy 0 = g (V (x, y), V (x, y) = V (x, y) x 0
37 31 (because λ preserves the inner products on T and T ). xy x y iv) This is obvious. ) v) Note that g (V (x, y), V (x, y) = 1 if x y so (ω ) = ω. x 0 0 Remark The sharp and flat in Definition are a special case of Definition when we choose σ and σ as a product of the Dirac delta function and V = V = V The pairing < ω, V > is not non-degenerate with respect to V. 3. The identity < V ( ), V 1 2 > (x, y) = g V (x, y), V (x, y) does not hold here. x The identity (V ) = V does not hold. 5. Even though the sharp and flat we define here do not satisfy all properties as in local calculus they are good enough in the sense that they guarantee the relations between operators in vector calculus as shown in following sections. 6. Here we use a fixed skew-symmetric vector field V to define sharp and flat. 0 Thus the dual of a 1-form is a symmetric vector field. We can also start with a fixed symmetric vector field V and define the dual of a 0 1-form as a skew-symmetric vector field using the same formulas above for sharp and flat. We obtain all the same properties listed. For example in R n with the trivial NL trivialization one can take V (x, y) = (x + y)/ x + y if x y (and 0 V (x, x) = 0) NL Vector Calculus In this section we will define all the usual operators in vector calculus using the invariant formulations as in local vector calculus. That is vector calculus operators are defined using only operators in exterior calculus such as sharp, flat,
38 32 exterior derivative, etc (in local calculus this means vector calculus operators can be written in coordinate-free notation). But first of all similar to Section 3.3 we would like to start with general definitions. Definition i) The NL gradient f of a 0-form f is a NL vector field defined as f(x, y) = M 3 σ g (x, y, x, y, z ) f(z ) λ x (V (x, y )) dx dy dz. x 0g Here σ : M 5 R is a given kernel, V is a given NL vector field. g 0g ii) The NL divergence div(v ) of a NL vector field V is a 0-form defined by div(v )(x) = M 5 σ d (x, x, y, x, y ( ) g x V (x, y ), λ x x (V (x, y ) )) dx dy dx dy. 0d Here σ d : M 5 R is a given kernel, V 0d is a given NL vector field. iii) The NL Laplacian f of a 0-form f is a 0-form defined by f(x) = M 3 σ L (x, x, y, z ) f(z ) dx dy dz Here σ : M 4 R is a given kernel. L Remark In (i) since f(x, y) T x we need to use the NL trivialization so that λ x x (V 0g (x, y )) T x. Similar reason for using NL trivialization in (ii). Just as in standard context, we would like to preserve similar relationships among operators in vector calculus. For example gradient and divergence are the adjoint of each other with respect to the inner product on Ω 0 NL (M) and an inner product on the space of NL vector fields which is induced from the inner product on
39 Ω 1 (M). Also the Laplacian should be written in terms of gradient and divergence NL and there should be a divergence theorem. We now show that the operators in vector calculus which are induced by the NL exterior calculus have special forms, which give us many standard relationships just as in local case. Theorem Suppose that the differential D and codifferential operators D correspond to the special operator T in Section 2.3 and the special inner product 33 in Section 2.4. Also suppose that the sharp and flat have special forms as in Definition i) If σ (x, y, x, y, z ) = (K(y, z ) K(x, z )) δ (x ) δ (y ) and V = V then g x y 0g 0 the gradient is induced by the exterior calculus, that is f = (Df). Here again δ is the Dirac delta function. ii) If σ (x, x, y, x, y ) = 2K(x, x ) W (x, y ) δ d 1 x (x ) δ y (y ) and V = V 0d 0 then the divergence is induced by the exterior calculus, that is div(v ) = D V. iii) If σ L (x, x, y, z ) = 2K(x, x )(K(y, z ) K(x, z ))W 1 (x, y ) then the Laplacian is induced by the exterior calculus, that is f = (D D + DD )f (the Laplace-Beltrami operator applies on a 0-form). Proof. i) (Df) ( (x, y) = M 2 K(x, x )K(y, y )(f(y ) f(x )) dx dy ) V (x, y) 0 ( = (K(y, z ) K(x, z ))f(z ) dz ) V (x, y) 0 M so f = (Df) if σ (x, y, x, y, z ) = (K(y, z ) K(x, z )) δ (x ) δ (y ) and g x y V = V. 0g 0
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationIntroduction to finite element exterior calculus
Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway Why finite element exterior calculus? Recall the de Rham complex on the form: R H 1 (Ω) grad H(curl, Ω) curl
More informationGEOMETRICAL TOOLS FOR PDES ON A SURFACE WITH ROTATING SHALLOW WATER EQUATIONS ON A SPHERE
GEOETRICAL TOOLS FOR PDES ON A SURFACE WITH ROTATING SHALLOW WATER EQUATIONS ON A SPHERE BIN CHENG Abstract. This is an excerpt from my paper with A. ahalov [1]. PDE theories with Riemannian geometry are
More informationLecture 4: Harmonic forms
Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationThe Hodge Star Operator
The Hodge Star Operator Rich Schwartz April 22, 2015 1 Basic Definitions We ll start out by defining the Hodge star operator as a map from k (R n ) to n k (R n ). Here k (R n ) denotes the vector space
More informationDIFFERENTIAL FORMS AND COHOMOLOGY
DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationHandout to Wu Characteristic Harvard math table talk Oliver Knill, 3/8/2016
Handout to Wu Characteristic Harvard math table talk Oliver Knill, 3/8/2016 Definitions Let G = (V, E) be a finite simple graph with vertex set V and edge set E. The f-vector v(g) = (v 0 (G), v 1 (G),...,
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationLecture 13. Differential forms
Lecture 13. Differential forms In the last few lectures we have seen how a connection can be used to differentiate tensors, and how the introduction of a Riemannian metric gives a canonical choice of connection.
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationDifferential Forms, Integration on Manifolds, and Stokes Theorem
Differential Forms, Integration on Manifolds, and Stokes Theorem Matthew D. Brown School of Mathematical and Statistical Sciences Arizona State University Tempe, Arizona 85287 matthewdbrown@asu.edu March
More informationMath 225B: Differential Geometry, Final
Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of
More informationTHE HODGE DECOMPOSITION
THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional
More informationMULTILINEAR ALGEBRA MCKENZIE LAMB
MULTILINEAR ALGEBRA MCKENZIE LAMB 1. Introduction This project consists of a rambling introduction to some basic notions in multilinear algebra. The central purpose will be to show that the div, grad,
More informationTowards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity
Towards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity Melvin Leok Mathematics, University of Michigan, Ann Arbor. Joint work with Mathieu Desbrun, Anil Hirani, and Jerrold
More informationSurvey on exterior algebra and differential forms
Survey on exterior algebra and differential forms Daniel Grieser 16. Mai 2013 Inhaltsverzeichnis 1 Exterior algebra for a vector space 1 1.1 Alternating forms, wedge and interior product.....................
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More information1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces.
1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces. The set of n-tuples of real numbers is denoted by R n. Suppose that a is a real number
More informationCONNECTIONS ON PRINCIPAL BUNDLES AND CLASSICAL ELECTROMAGNETISM
CONNECTIONS ON PRINCIPAL BUNDLES AND CLASSICAL ELECTROMAGNETISM APURV NAKADE Abstract. The goal is to understand how Maxwell s equations can be formulated using the language of connections. There will
More informationRIEMANN SURFACES. ω = ( f i (γ(t))γ i (t))dt.
RIEMANN SURFACES 6. Week 7: Differential forms. De Rham complex 6.1. Introduction. The notion of differential form is important for us for various reasons. First of all, one can integrate a k-form along
More informationLecture 5 - Lie Algebra Cohomology II
Lecture 5 - Lie Algebra Cohomology II January 28, 2013 1 Motivation: Left-invariant modules over a group Given a vector bundle F ξ G over G where G has a representation on F, a left G- action on ξ is a
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationFrom point cloud data to the continuum model of geometry
From point cloud data to the continuum model of geometry J. Harrison University of California, Berkeley July 22, 2007 Continuum model of geometry By the continuum model of geometry we refer to smooth manifolds
More informationSpectral Processing. Misha Kazhdan
Spectral Processing Misha Kazhdan [Taubin, 1995] A Signal Processing Approach to Fair Surface Design [Desbrun, et al., 1999] Implicit Fairing of Arbitrary Meshes [Vallet and Levy, 2008] Spectral Geometry
More informationCS 468 (Spring 2013) Discrete Differential Geometry
CS 468 (Spring 2013) Discrete Differential Geometry Lecture 13 13 May 2013 Tensors and Exterior Calculus Lecturer: Adrian Butscher Scribe: Cassidy Saenz 1 Vectors, Dual Vectors and Tensors 1.1 Inner Product
More informationMath 114: Course Summary
Math 114: Course Summary Rich Schwartz September 25, 2009 General Information: Math 114 is a course in real analysis. It is the second half of the undergraduate series in real analysis, M113-4. In M113,
More informationMaxwell s equations in Carnot groups
Maxwell s equations in Carnot groups B. Franchi (U. Bologna) INDAM Meeting on Geometric Control and sub-riemannian Geometry Cortona, May 21-25, 2012 in honor of Andrey Agrachev s 60th birthday Researches
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationTHE LOCAL THEORY OF ELLIPTIC OPERATORS AND THE HODGE THEOREM
THE LOCAL THEORY OF ELLIPTIC OPERATORS AND THE HODGE THEOREM BEN LOWE Abstract. In this paper, we develop the local theory of elliptic operators with a mind to proving the Hodge Decomposition Theorem.
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationPoisson configuration spaces, von Neumann algebras, and harmonic forms
J. of Nonlinear Math. Phys. Volume 11, Supplement (2004), 179 184 Bialowieza XXI, XXII Poisson configuration spaces, von Neumann algebras, and harmonic forms Alexei DALETSKII School of Computing and Technology
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Spring 2017 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u =
More informationCE-570 Advanced Structural Mechanics - Arun Prakash
Ch1-Intro Page 1 CE-570 Advanced Structural Mechanics - Arun Prakash The BIG Picture What is Mechanics? Mechanics is study of how things work: how anything works, how the world works! People ask: "Do you
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationSymplectic and Poisson Manifolds
Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to
More informationEnergy on fractals and related questions: about the use of differential 1-forms on the Sierpinski Gasket and other fractals
Energy on fractals and related questions: about the use of differential 1-forms on the Sierpinski Gasket and other fractals Part 2 A.Teplyaev University of Connecticut Rome, April May 2015 Main works to
More informationRiemannian geometry of surfaces
Riemannian geometry of surfaces In this note, we will learn how to make sense of the concepts of differential geometry on a surface M, which is not necessarily situated in R 3. This intrinsic approach
More informationElliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.
Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationThe Calabi Conjecture
The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationAn Introduction to Discrete Vector Calculus on Finite Networks
Grupo ARIDIS REU 010, UW Notes for the Lecture, June 010. An Introduction to Discrete ector Calculus on Finite Networks A. Carmona and A.M. Encinas We aim here at introducing the basic terminology and
More informationSyllabuses for Honor Courses. Algebra I & II
Syllabuses for Honor Courses Algebra I & II Algebra is a fundamental part of the language of mathematics. Algebraic methods are used in all areas of mathematics. We will fully develop all the key concepts.
More informationDIFFERENTIAL FORMS, SPINORS AND BOUNDED CURVATURE COLLAPSE. John Lott. University of Michigan. November, 2000
DIFFERENTIAL FORMS, SPINORS AND BOUNDED CURVATURE COLLAPSE John Lott University of Michigan November, 2000 From preprints Collapsing and the Differential Form Laplacian On the Spectrum of a Finite-Volume
More informationOn the spectrum of the Hodge Laplacian and the John ellipsoid
Banff, July 203 On the spectrum of the Hodge Laplacian and the John ellipsoid Alessandro Savo, Sapienza Università di Roma We give upper and lower bounds for the first eigenvalue of the Hodge Laplacian
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationIn collaboration with Mathieu Desbrun, Anil N. Hirani, and Jerrold E. Marsden. Abstract
73 Chapter 3 Discrete Exterior Calculus In collaboration with Mathieu Desbrun, Anil N. Hirani, and Jerrold E. Marsden. Abstract We present a theory and applications of discrete exterior calculus on simplicial
More informationDifferential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.
Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product
More informationTopic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016
Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces
More informationSheaf theory August 23, 2016
Sheaf theory August 23, 216 Chapter 1 The statement of de Rham s theorem Before doing anything fancy, let s start at the beginning. Let U R 3 be an open set. In calculus class, we learn about operations
More informationFOUNDATIONS OF COMPUTATIONAL GEOMETRIC MECHANICS
FOUNDATIONS OF COMPUTATIONAL GEOMETRIC MECHANICS MELVIN LEOK Received the SIAM Student Paper Prize and the Leslie Fox Prize in Numerical Analysis (second prize). Abstract. Geometric mechanics involves
More informationKlaus Janich. Vector Analysis. Translated by Leslie Kay. With 108 Illustrations. Springer
Klaus Janich Vector Analysis Translated by Leslie Kay With 108 Illustrations Springer Preface to the English Edition Preface to the First German Edition Differentiable Manifolds 1 1.1 The Concept of a
More informationSubmanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition.
le 27 AIPEI CHENNAI TAIPEI - Series in Pure Mathematics Volume 27 Total Mean Curvature and Submanifolds of Finite Type Second Edition Bang-Yen Chen Michigan State University, USA World Scientific NEW JERSEY
More informationA NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS
Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS QIANG DU Department
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationDISCRETE EXTERIOR CALCULUS
DISCRETE EXTERIOR CALCULUS MATHIEU DESBRUN, ANIL N. HIRANI, MELVIN LEOK, AND JERROLD E. MARSDEN Abstract. We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary
More informationFormality of Kähler manifolds
Formality of Kähler manifolds Aron Heleodoro February 24, 2015 In this talk of the seminar we like to understand the proof of Deligne, Griffiths, Morgan and Sullivan [DGMS75] of the formality of Kähler
More information4.7 The Levi-Civita connection and parallel transport
Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves
More informationDirectional Field. Xiao-Ming Fu
Directional Field Xiao-Ming Fu Outlines Introduction Discretization Representation Objectives and Constraints Outlines Introduction Discretization Representation Objectives and Constraints Definition Spatially-varying
More informationTHE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES
THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES Denis Bell 1 Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu This
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationCohomology of Harmonic Forms on Riemannian Manifolds With Boundary
Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary Sylvain Cappell, Dennis DeTurck, Herman Gluck, and Edward Y. Miller 1. Introduction To Julius Shaneson on the occasion of his 60th birthday
More informationPoisson Equation on Closed Manifolds
Poisson Equation on Closed anifolds Andrew acdougall December 15, 2011 1 Introduction The purpose of this project is to investigate the poisson equation φ = ρ on closed manifolds (compact manifolds without
More informationarxiv: v2 [math.na] 8 Sep 2015
OMPLEXES OF DISRETE DISTRIBUTIONAL DIFFERENTIAL FORMS AND THEIR HOMOLOGY THEORY MARTIN WERNER LIHT arxiv:1410.6354v2 [math.na] 8 Sep 2015 Abstract. omplexes of discrete distributional differential forms
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 29 July 2014 Geometric Analysis Seminar Beijing International Center for
More informationInvariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents
More informationAnalysis-3 lecture schemes
Analysis-3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space
More information1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces
More informationAdvanced Course: Transversal Dirac operators on distributions, foliations, and G-manifolds. Ken Richardson and coauthors
Advanced Course: Transversal Dirac operators on distributions, foliations, and G-manifolds Ken Richardson and coauthors Universitat Autònoma de Barcelona May 3-7, 2010 Universitat Autònoma de Barcelona
More informationSeminar on Vector Field Analysis on Surfaces
Seminar on Vector Field Analysis on Surfaces 236629 1 Last time Intro Cool stuff VFs on 2D Euclidean Domains Arrows on the plane Div, curl and all that Helmholtz decomposition 2 today Arrows on surfaces
More informationHODGE THEORY AND ELLIPTIC REGULARITY
HODGE THEORY AND ELLIPTIC REGULARITY JACKSON HANCE Abstract. The central goal of this paper is a proof of the Hodge decomposition of the derham complex for compact Riemannian manifolds. Along the way,
More informationLecture 17. Higher boundary regularity. April 15 th, We extend our results to include the boundary. Let u C 2 (Ω) C 0 ( Ω) be a solution of
Lecture 7 April 5 th, 004 Higher boundary regularity We extend our results to include the boundary. Higher a priori regularity upto the boundary. Let u C () C 0 ( ) be a solution of Lu = f on, u = ϕ on.
More informationDierential geometry for Physicists
Dierential geometry for Physicists (What we discussed in the course of lectures) Marián Fecko Comenius University, Bratislava Syllabus of lectures delivered at University of Regensburg in June and July
More informationarxiv: v1 [math-ph] 12 Dec 2007
arxiv:0712.2030v1 [math-ph] 12 Dec 2007 Green function for a two-dimensional discrete Laplace-Beltrami operator Volodymyr Sushch Koszalin University of Technology Sniadeckich 2, 75-453 Koszalin, Poland
More informationLoos Symmetric Cones. Jimmie Lawson Louisiana State University. July, 2018
Louisiana State University July, 2018 Dedication I would like to dedicate this talk to Joachim Hilgert, whose 60th birthday we celebrate at this conference and with whom I researched and wrote a big blue
More informationIntroduction to the Baum-Connes conjecture
Introduction to the Baum-Connes conjecture Nigel Higson, John Roe PSU NCGOA07 Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 1 / 15 History of the BC conjecture Lecture
More informationClassical differential geometry of two-dimensional surfaces
Classical differential geometry of two-dimensional surfaces 1 Basic definitions This section gives an overview of the basic notions of differential geometry for twodimensional surfaces. It follows mainly
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationChoice of Riemannian Metrics for Rigid Body Kinematics
Choice of Riemannian Metrics for Rigid Body Kinematics Miloš Žefran1, Vijay Kumar 1 and Christopher Croke 2 1 General Robotics and Active Sensory Perception (GRASP) Laboratory 2 Department of Mathematics
More informationHOMOGENEOUS AND INHOMOGENEOUS MAXWELL S EQUATIONS IN TERMS OF HODGE STAR OPERATOR
GANIT J. Bangladesh Math. Soc. (ISSN 166-3694) 37 (217) 15-27 HOMOGENEOUS AND INHOMOGENEOUS MAXWELL S EQUATIONS IN TERMS OF HODGE STAR OPERATOR Zakir Hossine 1,* and Md. Showkat Ali 2 1 Department of Mathematics,
More informationHomogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky
Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey
More informationν(u, v) = N(u, v) G(r(u, v)) E r(u,v) 3.
5. The Gauss Curvature Beyond doubt, the notion of Gauss curvature is of paramount importance in differential geometry. Recall two lessons we have learned so far about this notion: first, the presence
More informationLecture I: Constrained Hamiltonian systems
Lecture I: Constrained Hamiltonian systems (Courses in canonical gravity) Yaser Tavakoli December 15, 2014 1 Introduction In canonical formulation of general relativity, geometry of space-time is given
More informationFINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS
FINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS GILES AUCHMUTY AND JAMES C. ALEXANDER Abstract. This paper describes the existence and representation of certain finite energy (L 2 -) solutions of
More informationRIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY
RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C
More informationTWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY
TWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY Andrei Moroianu CNRS - Ecole Polytechnique Palaiseau Prague, September 1 st, 2004 joint work with Uwe Semmelmann Plan of the talk Algebraic preliminaries
More information