NONLOCAL EXTERIOR CALCULUS ON RIEMANNIAN MANIFOLDS

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

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.

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.

iv Table of Contents Acknowledgments................................ vi Chapter 1. Introduction............................ 1 1.1 Local Exterior Calculus and Finite Element Exterior Calculus 2 1.2 Discrete Exterior Calculus.................... 3 1.3 Existing Works in Nonlocal Vector Calculus.......... 4 1.4 Hodge Theory on Metric Spaces................. 5 1.5 Details of this Dissertation.................... 7 Chapter 2. Nonlocal forms.......................... 10 2.1 Oriented Simplices and Tuples.................. 10 2.2 Nonlocal Forms.......................... 11 2.3 Nonlocal Exterior Derivative (D)................ 12 2.4 Codifferential Operator (D )................... 17 2.5 NL Laplace-Beltrami Operator ( )............... 21 2.6 NL Hodge Operator ( )...................... 21 2.7 Nonlocal Wedge Product ).................. nl 24 Chapter 3. Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus............... 25 3.1 Nonlocal Trivializations (λ)................... 25 3.2 Vector Fields........................... 27 3.3 NL Sharp ( ) and Flat ( ) Operators.............. 27 3.4 NL Vector Calculus........................ 31

3.5 Some special cases of NL vector calculus operators....... 37 v Chapter 4. Relationships to Local Geometry, Classical Vector Calculus and Discrete Vector Calculus..................... 43 4.1 Relationship between NL geometry and local geometry.... 43 4.2 Relationship between NL Vector Calculus and the Local One. 45 4.3 Relationship between NL Laplacian and discrete Laplacian.. 46 Chapter 5. NL Hodge Theory......................... 50 5.1 Hodge Theory for the Discrete Derivative............ 50 5.2 Hodge Theory for the Nonlocal Exterior Derivative...... 63 Chapter 6. Another Model for Nonlocal Exterior Calculus in R n..... 67 6.1 Nonlocal Forms.......................... 67 6.2 Nonlocal Exterior Derivative (D)................ 68 6.3 Codifferential Operator (D )................... 69 6.4 NL Laplace-Beltrami Operator (revisit)............. 70 6.5 Hodge Operator ( )........................ 70 6.6 Sharp ( ) and Flat ( ) Operators................. 73 6.7 Vector Calculus (revisit)..................... 75 6.8 NL Wedge Product (Λ nl ).................... 76 Chapter 7. Ongoing and Future Works.................... 78 7.1 Ongoing Works.......................... 78 7.2 Future Works........................... 78 References.................................... 80

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.

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.

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).

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.

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

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

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

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

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

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.

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

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 2.1.1 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 2.1.2 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 2.2.1 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 2.2.2 1. 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

12 opposite sign. We refer to [17] (p. 29-30) 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 2.3.1 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 2.3.2 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).

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 2.3.3 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 2.3.4 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

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 2.3.5 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

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

16 (i = 0, 1,...). Thus T satisfies the identity (2.4) and it belongs to T NL (M). Remark 2.3.6 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.

17 2.4 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 2.4.1 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 2.4.2 1. 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. 85-86) for the definition of the codifferential operator in local exterior calculus. 3. Because D 2 = 0 we also have (D ) 2 = 0.

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 1. 1 1 The other cases are similar to examples 2 and 3 in Remark 2.3.6 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. 1010-1011) for more discussion about W 1 in R n. For a general manifold M one can use the heat kernel as in Remark 2.3.6 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).

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 2.4.3 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 2.4.4 1. 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)

Proposition 2.4.5 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 2.4.6 One can also define the operator D using the formula (2.10) then prove that it is formally adjoint to D.

21 2.5 NL Laplace-Beltrami Operator ( ) Definition 2.5.1 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.

Lemma 2.6.1 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

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 2.6.2 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).

Here due to the nonlocality there is no guarantee that the NL Hodge still possesses these properties. 24 2.7 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. 74-75). Definition 2.7.1 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. 17-21 and p. 49) for the proof. Remark 2.7.2 The wedge product defined here only works for the discrete derivative d.

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 3.1.1 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 3.1.2 1. We refer to [26] for more discussions about NL trivialization(s). The author in that manuscript only imposes condition (i) for a nonlocal trivialization.

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

Riemannian geometry. Some of them will be discussed in subsequent sections. We refer to [26] for more details on NL differential geometry discussed there. 27 3.2 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 3.2.1 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 3.2.2 1. 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

a general definition for each of these operators and then we will use a specific one for computation and application purposes. 28 Definition 3.3.1 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 3.3.2 1. 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

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 0 0 0 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), 2 1 1 +x2 2 if x 0 1 Now we define

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 0 0 0 Definition 3.3.3 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 3.3.4 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

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 3.3.5 1. The sharp and flat in Definition 3.3.3 are a special case of Definition 3.3.1 when we choose σ and σ as a product of the Dirac delta function and V = V = V. 0 0 0 2. 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 1 2 4. 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). 0 3.4 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,

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 3.4.1 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 3.4.2 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

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 3.4.3 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 3.3.3. 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