# 1. Geometry of the unit tangent bundle

Size: px
Start display at page:

Transcription

1 1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g Notations Given a local chart (U, φ) on M, we will denote by (x i ) 1 i n the local coordinates and we write in these coordinates n g = g ij dx i dx j, ( where g ij (x) = g, x i i,j=1 ). We denote by (g ij ) the coefficient of the x j inverse matrix. Given x M, the norm of v T x M is given by v x = g x (v, v) 1/2, which will sometimes be denoted v, v 1/2 x. In the following, we will often drop the index x. Note that if we are given other coordinates (y j ), then one can check that in these new coordinates : n (1.1) g = i,j,k,l=1 g ij x i y k x j y l dy k dy l We define the musical isomorphism at x M by : T xm Tx M v v = g(v, ) This is an isomorphism between the two vector bundles T M and T M since they are equidimensional and g is symmetric definite positive, thus non-degenerate. Given an orthonormal basis (e i ) of T x M, we will denote by (e i ) the dual basis of Tx M which is, in other words, the image of the basis (e i ) by the musical isomorphism. If E is a vector bundle over M, then the projection will be denoted π : E M. We denote by Γ(M, E) the set of smooth sections of E. Γ(M) denotes Γ(M, T M), the set of vector fields. We will denote by Γ(M, m S T M) the set of smooth symmetric covariant m-tensors on M. On the cotangent bundle T M, we denote by ω the canonical symplectic form, which we write in coordinates ω = n i=1 dpi dx i. We recall that it is obtained as the differential of the canonical 1-form λ Ω 1 (T M), defined intrinsically as λ (x,p) (ξ) = p ( dπ (x,p) (ξ) ),

2 2 for a point (x, p) T M, ξ T (x,p) T M and where π : T M M denotes the projection. Given a point x M, if (e i ) is an orthonormal basis of T x M, we define dvol = e 1... e n. In local coordinates, it is given by the formula : (1.2) dvol = det(g ij )dx 1... dx n We recall that the Laplacian in local coordinates is given by : n 1 ( ) (1.3) f = i det gg ij j f det g i,j=1 We denote by the Levi-Civita connection on T M. In local coordinates, the Christoffel symbols are defined such that : n (1.4) = Γ k ij x i x j x k k=1 They are given by the Koszul formula : (1.5) Γ k ij = 1 n ( g kl gil + g jl g ) ij 2 x j x i x l l=1 We denote the torsion tensor T, which is defined as : T (X, Y ) = X Y Y X [X, Y ] The curvature tensor is denoted by F and defined as : (1.6) F (X, Y )Z = X Y Z Y X Z [X,Y ] Z In particular, we clearly have F (X, Y ) = F (Y, X). We recall that the Levi-Civita connection is the unique torsion-free and g-metric connection, namely it satisfies for any X, Y, Z Γ(M) : (1.7) T (X, Y ) = X Y Y X [X, Y ] = 0 (1.8) Z (g(x, Y )) = g( Z X, Y ) + g(x, Z Y ) The sectional curvature K is given at x by K x (e 1, e 2 ) = F (e 1, e 2 )e 1, e 2, where e 1, e 2 T x M are orthogonal. In particular, in the case of a surface, which is what we will be mostly interested in, the sectional curvature is a real number referred to as the Gaussian curvature (or simply the curvature if the context is not ambiguous).

3 Structure of SM The horizontal distribution Let us first state some very general results. We consider a vector bundle π : E M of rank r with a connection. Given a path γ : I M such that γ(0) = x, γ(0) = X T x M and an initial value s 0 E x = π 1 ({x}), there exists a unique lift of γ to a path s : I E it is the parallel transport of s 0 along γ such that s(0) = s 0, π(s) = γ and (1.9) γ s = 0 Now, we can associate to X the vector d (1.10) X = dt s(t) T s0 E t=0 One can check that X is well defined and only depends on the choice of (γ(0), γ(0)). It is a linear application T x M T s0 E which we denote by θ : X X. In local coordinates, if we write X = X i, s 0 = s j e j, then x i the parallel transport equation (1.9) yield to : (1.11) θ(x) = X i Γ k x ijx i s j i s k We define the horizontal distribution on (E, ) at s 0 E as : (1.12) H s 0 = Span(θ(X), X T π (s 0 )) T s0 E In the following, we will drop the notation for H s0, but we insist on the fact that H s0 is entirely determined by the choice of. Actually, one can check that choosing a connection is strictly equivalent to choosing a smooth distribution of vector spaces H T E. Note that we have the following theorem : Theorem 1.1. The following assertions are equivalent : The distribution s 0 H s0 is integrable. The connection is flat, i.e. T vanishes. The holonomy i.e. the parallel transport of a section along a closed loop is invariant by homotopy of this loop. Since θ is clearly injective by definition, it defines an isomorphism between T π(s0) H s0. We define the vertical distribution on (E, ) at s 0 as V s0 = ker dπ s0 = E π(s0) T s0 E. Thanks to (1.9), it is easy to check that θ is exactly the inverse of the differential dπ s0 restricted to the subspace H s0 of T s0 E. We therefore have the splitting : (1.13) T s0 E = H s0 V s0

4 4 X i Given a vector ξ T s0 E, which we write in local coordinates ξ = + Y k, (1.11) shows that it is in H s x i s 0 if and only if k (1.14) Y k + Γ k ijx i s j = 0, for all 1 k r, and it is in V s0 if and only if X i = 0 for all 1 i n The Sasaki metric We now apply the previous formalism to the particular case when E = T M and construct a canonical metric, on T M, called the Sasaki metric. If is the Levi-Civita connection on T M, then it automatically determines a splitting like in (1.13). In this case, there also exists a canonical application K : T (x,v) T M T x M whose kernel is exactly H (x,v). Somehow, it can be seen as a complementary application to dπ. Given ξ T (x,v) T M, we can consider a local curve c in T M such that c(0) = (x, v) and ċ(0) = ξ. We can write c(t) = (γ(t), Z(t)), where Z is a vector field along the curve γ. We define : (1.15) K (x,v) (ξ) = γ Z(0) One can easily check that the application K now defines an isomorphism between V (x,v) and T x M and its kernel is precisely H (x,v), the horizontal distribution. Definition 1.2. Given a point (x, v) T M and ξ, η T (x,v) T M, we set : If ξ, η are vertical, namely ξ, η V (x,v) = T x M, then ξ, η := K(ξ), K(η), If ξ, η are horizontal, then ξ, η := dπ (x,v) (ξ), dπ (x,v) η, V (x,v) and H (x,v) are orthogonal for,. Note that the last line induces in particular that the decomposition (1.13) is orthogonal for the Sasaki metric. In the rest of this paragraph, we explain how to recover the Sasaki metric from the symplectic point of view. We recall that λ is the canonical 1-form defined on T M and introduced in the previous section. It can be pulled

5 5 back via the musical isomorphism to get α = λ Ω 1 (T M). Then α (x,v) (ξ) = λ (x,v )(d (x,v) (ξ)) = v ( dπ (x,v ) d (x,v) (ξ) ) ( ) = v d (π ) (x,v) (ξ) = v, dπ (x,v) (ξ) = Z, ξ, for some vector field Z, according to Riesz representation theorem. We claim that Z is actually the geodesic vector field X. Indeed, first observe that X is a horizontal vector field (i.e. X(x, v) H (x,v) ), which is an immediate consequence of the expressions in local coordinates (??) and (1.14). In particular, in local coordinates, expression (1.11) yield to : (1.16) dπ (x,v) (X(x, v)) = v i x i In order to prove that Z = X, we just have to check that the expressions v, dπ (x,v) (ξ) and X, ξ agree when ξ runs through a basis of T (x,v) T M. If ξ is in V (x,v) (that is ξ is vertical), this is immediate since dπ(ξ) = 0 and X is horizontal. If ξ = ξ i = Γ k ij x vj, then dπ(ξ i ) = by (1.11) i v k x i and v, dπ (x,v) (ξ) = v i. But by definition of the Sasaki metric, since both X and ξ i are horizontal : We therefore obtain the equality : X, ξ i = dπ(x), dπ(ξ i ) = v i (1.17) α (x,v) (ξ) = X, ξ In other words, α = X, where : T T M T T M is the musical isomorphism given by the Sasaki metric. Remark 1.3. Behind this is hidden the fact that SM (the unit tangent bundle), and therefore S M, both have the structure of a contact manifold, but we will not give further details about this Surface theory Isothermal coordinates In this paragraph, we introduce isothermal coordinates, which will be widely used is the following.

6 6 Definition 1.4. Let (M, g) be an n-dimensional Riemannian manifold. Isothermal coordinates are local coordinates such that the metric can be written g = e 2λ (dx dx 2 n), where λ is a smooth function. In dimension n 3, in a neighborhood of a point, isothermal coordinates may not exist. However, in dimension n = 2, we have the Theorem 1.5. Let (M, g) be a Riemannian surface and x M. There exists isothermal coordinates in a neighborhood of x. We provide a proof of this theorem in Appendix?? based on the resolution of a Dirichlet problem. The existence of isothermal coordinates on a surface is closely link to the existence of a Riemann structure (or holomorphic structure) on the surface, that is a covering by charts {U, ϕ} such that the transition maps are all holomorphic, as explained in the Appendix. A complex structure on M is an endomorphism J End(T M) such that J 2 = id. In particular, the data of a given conformal class together with an orientation of the manifold is equivalent to that of a complex structure. The Koszul formula allows to compute the Christoffel symbols in the isothermal coordinates. We obtain : Γ 1 ij j = 1 j = 2 i = 1 1 λ 2 λ i = 2 2 λ 1 λ Γ 2 ij j = 1 j = 2 i = 1 2 λ 1 λ i = 2 1 λ 2 λ Note that in this case, the curvature of M has a rather simple expression : (1.18) K = e 2λ λ The moving frame We now assume that M is a surface and is the Levi-Civita connection on M, associated to the Riemannian metric g. Instead of studying the tangent bundle T M, we will study the unit tangent bundle SM. Note, that since we will mostly be interested in studying properties of the geodesic flow, this will not restrict our considerations as far as geodesics are covered at constant speed (and one can always assume it is arc-length parametrized, up to a preliminary reparametrization). SM can be seen as a subriemannian manifold of T M. In the following, we may identify the tangent space T x M at a point with the complex plane

7 7 C and therefore work with complex coordinates. In other words, this simply means that we see M as a Riemannian surface, endowed with a complex structure i, and i simply acts on tangent vectors by π/2-rotating them. S x will denote the restriction of T x M to the unit circle. Given a point (x, v) SM, the rotation r θ : S x S x defined in coordinates by r θ (x, v) = (x, e iθ v) is a flow on SM which provides an infinitesimal generator V (x, v), spanning a direction of T (x,v) SM. The horizontal distribution H (x,v) is unchanged and since SM is submanifold of T M of dimension 3, one gets the orthogonal splitting : T (x,v) SM = Span(V (x, v)) H (x,v) Moreover, thanks to the expression (1.16), one clearly sees that the vector field X is unitary on SM. This is also the case for the vector field V. In order to obtain a orthonormal basis of T (x,v) SM at a point (x, v) SM, we need to provide a third vector field H H (x,v), unitary and orthogonal to X. It is chosen so that {X, H, V } is positively oriented. We call this basis the moving frame on SM. We can give explicit formulas for this vector fields in isothermal coordinates. In these coordinates, in a neighborhood U of a point x M, we write g = e 2λ(x1,x2) (dx dx 2 2). The local coordinates induced on T M U are denoted by (x 1, x 2, v 1, v 2 ). Thus, we can consider local coordinates on SM U, given by : (x 1, x 2, θ) (x 1, x 2, e λ cos θ, e λ sin θ) Thanks to the Koszul formula (1.5), the Christoffel symbols can be computed explicitly. The expression (??) then yield to : (1.19) ( X(x, θ) = e λ cos θ + sin θ ( + λ sin θ + λ ) ) cos θ x 1 x 2 x 1 x 2 θ V (x, θ) is simply given by : (1.20) V (x, θ) = θ Eventually, H(x, θ) can be obtained as a normal vector to the plane spanned by the vectors {X, V }, or as a positive rotation of X in the plane H x,v of angle π/2. We get : (1.21) ( H(x, θ) = e λ sin θ + cos θ ( λ sin θ + λ ) ) cos θ x 1 x 1 x 1 x 2 θ

8 8 From these formulas, we can derive the Cartan structural equations. In particular, using the expression (1.18) for the curvature, one gets : (1.22) [X, H] = K V (1.23) [V, X] = H (1.24) [H, V ] = X T (x;v) SM V(x;v) H(x; v) X(x; v) H (x;v) K 1 (v) 0 π 1 (fxg) π iv T x M x v M Figure 1.1. The unit tangent bundle We can study this basis from the dual point of view. Via the musical isomorphism defined thanks to the Sasaki metric, we obtain a dual basis {α, β, ψ} on T SM (note that the 1-form α has already been introduced in the previous section). Let us give a short description of this basis. Given ξ T (x,v) SM, since dπ (x,v) (X) = v, dπ (x,v) (H) = iv, K (x,v) (V ) = iv, we have : α (x,v) (ξ) = X, ξ = v, dπ (x,v) (ξ), β (x,v) (ξ) = H, ξ = iv, dπ (x,v) (ξ), ψ (x,v) (ξ) = V, ξ = iv, K (x,v) (ξ).

9 9 The kernel of ψ is precisely the horizontal distribution H. The dual formulation (1) of the Cartan structural equations is : (1.25) α β = K dψ (1.26) ψ α = dβ (1.27) β ψ = dα Remark 1.6. For the reader s convenience, let us just quote here the inversion formulas : = e λ (cos θ X sin θ H) λ V x 1 x 2 x 2 = e λ (sin θ X + cos θ H) + λ x 1 V Canonical volume form on SM As we mentioned before, the metric g induces a canonical volume form on the surface M which can be written in isothermal coordinates, thanks to formula (1.2) : (1.28) dvol = e 2λ dx 1 dx 2 We can also define a canonical volume form on SM, denoted by Θ, as the canonical volume form induced by the Sasaki metric. It is called the Liouville measure on SM. Since the dual basis {α, β, ψ} is orthonormal for the Sasaki metric, then one simply has : (1.29) Θ := α β ψ From the definition (1.29), and the Cartan structural equations, one can prove that the fields X, H and V preserve the volume form Θ. Indeed, let us prove it for X. If L X denotes the Lie derivative with respect to X, then one has : (L X Θ) (X, H, V ) = L X (Θ(X, H, V )) Θ([X, X], H, V ) Θ(X, [X, H], V ) Θ(X, H, [X, V ]) Since Θ(X, H, V ) = 1, the first term is zero and using Cartan structural equations, the three other terms are zero. So L X Θ is zero, that is, X preserves the volume-form Θ. 1. Dual in the sense that we apply d to the Cartan structural equations on the vector fields.

10 10 We also have Θ = π (dvol) dθ. Indeed, since Θ and π (dvol) dθ are both volume forms, we know that they may differ by a multiplicative function. But Θ(X, H, V ) = 1 and π (dvol) dθ(x, H, V ) = dvol(dπ(x), dπ(h)) = 1. Given u C (SM), this yields to the integration formula over the fibers : ( ) (1.30) u Θ = u SMπ(p) (p) dθ(p) dvol(π(p)) SM M SM π(p) In particular : (1.31) vol(sm) = 2π vol(m) Remark 1.7. The volume form Θ can also be seen as the contact form α (dα) n 1. Here, since n = 2, we simply obtain α dα = α β ψ = Θ by the dual formulation of the Cartan structural equation (1.27). Remark 1.8. The construction of the Sasaki metric on the tangent bundle and the canonical volume form on the unit tangent bundle is not specific to dimension 2 and can easily be generalized to greater dimensions. Actually, the unit tangent bundle is locally a product space U S n 1 and the Liouville measure on SM is nothing but the product measure dvol ds where ds denotes the euclidean volume form on S n 1. BIBLIOGRAPHY [1] S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Springer, [2] M. E. Taylor, Partial Differential Equations I & II, Applied Mathematical Sciences, Volume 116, Springer, Thibault LEFEUVRE École Polytechnique Palaiseau France Current address: Université de Paris VI Faculté des sciences de Jussieu Paris

### CALCULUS 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.

### A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström

A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal

### Exercises 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

### DIFFERENTIAL GEOMETRY. LECTURE 12-13,

DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of

### Complex line bundles. Chapter Connections of line bundle. Consider a complex line bundle L M. For any integer k N, let

Chapter 1 Complex line bundles 1.1 Connections of line bundle Consider a complex line bundle L M. For any integer k N, let be the space of k-forms with values in L. Ω k (M, L) = C (M, L k (T M)) Definition

### Smooth Dynamics 2. Problem Set Nr. 1. Instructor: Submitted by: Prof. Wilkinson Clark Butler. University of Chicago Winter 2013

Smooth Dynamics 2 Problem Set Nr. 1 University of Chicago Winter 2013 Instructor: Submitted by: Prof. Wilkinson Clark Butler Problem 1 Let M be a Riemannian manifold with metric, and Levi-Civita connection.

### Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields

Chapter 15 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields The goal of this chapter is to understand the behavior of isometries and local isometries, in particular

### 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

### Some aspects of the Geodesic flow

Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques

### INTRO TO SUBRIEMANNIAN GEOMETRY

INTRO TO SUBRIEMANNIAN GEOMETRY 1. Introduction to subriemannian geometry A lot of this tal is inspired by the paper by Ines Kath and Oliver Ungermann on the arxiv, see [3] as well as [1]. Let M be a smooth

### THEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009

[under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was

### Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18

Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results

### WARPED PRODUCTS PETER PETERSEN

WARPED PRODUCTS PETER PETERSEN. Definitions We shall define as few concepts as possible. A tangent vector always has the local coordinate expansion v dx i (v) and a function the differential df f dxi We

### Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields

Chapter 16 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields 16.1 Isometries and Local Isometries Recall that a local isometry between two Riemannian manifolds M

### 4.7 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

### Conification of Kähler and hyper-kähler manifolds and supergr

Conification of Kähler and hyper-kähler manifolds and supergravity c-map Masaryk University, Brno, Czech Republic and Institute for Information Transmission Problems, Moscow, Russia Villasimius, September

### Homogeneous 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

### class # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS

class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]

### 1 First and second variational formulas for area

1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### From holonomy reductions of Cartan geometries to geometric compactifications

From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science

### DIFFERENTIAL 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,

### BERGMAN 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

### As 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

### Reduction of Homogeneous Riemannian structures

Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad

### Comparison for infinitesimal automorphisms. of parabolic geometries

Comparison techniques for infinitesimal automorphisms of parabolic geometries University of Vienna Faculty of Mathematics Srni, January 2012 This talk reports on joint work in progress with Karin Melnick

### Differential Geometry Exercises

Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the

### SOME EXERCISES IN CHARACTERISTIC CLASSES

SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined

### M4P52 Manifolds, 2016 Problem Sheet 1

Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

### Exercise 1 (Formula for connection 1-forms) Using the first structure equation, show that

1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral

### UNIVERSITY OF DUBLIN

UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429

### Integration of non linear conservation laws?

Integration of non linear conservation laws? Frédéric Hélein, Institut Mathématique de Jussieu, Paris 7 Advances in Surface Theory, Leicester, June 13, 2013 Harmonic maps Let (M, g) be an oriented Riemannian

### Lecture 8. Connections

Lecture 8. Connections This lecture introduces connections, which are the machinery required to allow differentiation of vector fields. 8.1 Differentiating vector fields. The idea of differentiating vector

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.1. SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM BY N. PAPAGHIUC 1

### Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

### NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES

NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES SHILIN U ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of

### Topology III Home Assignment 8 Subhadip Chowdhury

Topology Home Assignment 8 Subhadip Chowdhury Problem 1 Suppose {X 1,..., X n } is an orthonormal basis of T x M for some point x M. Then by definition, the volume form Ω is defined to be the n form such

### William 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

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

### Introduction to Differential Geometry

More about Introduction to Differential Geometry Lecture 7 of 10: Dominic Joyce, Oxford University October 2018 EPSRC CDT in Partial Differential Equations foundation module. These slides available at

### ADAPTED COMPLEX STRUCTURES AND GEOMETRIC QUANTIZATION

R. Szőke Nagoya Math. J. Vol. 154 1999), 171 183 ADAPTED COMPLEX STRUCTURES AND GEOMETRIC QUANTIZATION RÓBERT SZŐKE Abstract. A compact Riemannian symmetric space admits a canonical complexification. This

### Infinitesimal Einstein Deformations. Kähler Manifolds

on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds

### Geometry 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

### CHAPTER 1 PRELIMINARIES

CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable

### Solvable Lie groups and the shear construction

Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016 1 Swann s twist 2 The shear construction The

### THE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW

THE LONGITUDINAL KAM-COCYCLE OF A MAGNETIC FLOW GABRIEL P. PATERNAIN Abstract. Let M be a closed oriented surface of negative Gaussian curvature and let Ω be a non-exact 2-form. Let λ be a small positive

### LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction

LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural

### PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH

PHYS 4390: GENERAL RELATIVITY NON-COORDINATE BASIS APPROACH 1. Differential Forms To start our discussion, we will define a special class of type (0,r) tensors: Definition 1.1. A differential form of order

### LECTURE 10: THE PARALLEL TRANSPORT

LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be

### Geometry 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

### The spectral action for Dirac operators with torsion

The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011 1 Torsion Geometry, Einstein-Cartan-Theory

### Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups

Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,

### zi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn

The Complex Projective Space Definition. Complex projective n-space, denoted by CP n, is defined to be the set of 1-dimensional complex-linear subspaces of C n+1, with the quotient topology inherited from

### Riemannian geometry of the twistor space of a symplectic manifold

Riemannian geometry of the twistor space of a symplectic manifold R. Albuquerque rpa@uevora.pt Departamento de Matemática, Universidade de Évora Évora, Portugal September 004 0.1 The metric In this short

### Clifford Algebras and Spin Groups

Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of

### Many of the exercises are taken from the books referred at the end of the document.

Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the

### Theorem. In terms of the coordinate frame, the Levi-Civita connection is given by:

THE LEVI-CIVITA CONNECTION FOR THE POINCARÉ METRIC We denote complex numbers z = x + yi C where x, y R. Let H 2 denote the upper half-plane with the Poincaré metric: {x + iy x, y R, y > 0} g = dz 2 y 2

### Brownian Motion on Manifold

Brownian Motion on Manifold QI FENG Purdue University feng71@purdue.edu August 31, 2014 QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26 Overview 1 Extrinsic construction

### Hyperkähler 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

### Let F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.

Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages 59 64 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract.

### Lecture Notes a posteriori for Math 201

Lecture Notes a posteriori for Math 201 Jeremy Kahn September 22, 2011 1 Tuesday, September 13 We defined the tangent space T p M of a manifold at a point p, and the tangent bundle T M. Zev Choroles gave

### Riemannian Lie Subalgebroid

Université d Agadez; Faculté des Sciences, Agadez, Niger Riemannian Lie Subalgebroid Mahamane Saminou ALI & Mouhamadou HASSIROU 2 2 Université Abdou Moumouni; Faculté des Sciences et Techniques, Niamey,

### 5.2 The Levi-Civita Connection on Surfaces. 1 Parallel transport of vector fields on a surface M

5.2 The Levi-Civita Connection on Surfaces In this section, we define the parallel transport of vector fields on a surface M, and then we introduce the concept of the Levi-Civita connection, which is also

### Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

### The 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

### AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS. metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone (1) C = s>0

AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS JOHN C. LOFTIN 1. Introduction In this note, we introduce a straightforward correspondence between some natural affine Kähler metrics on convex cones and natural

### Some topics in sub-riemannian geometry

Some topics in sub-riemannian geometry Luca Rizzi CNRS, Institut Fourier Mathematical Colloquium Universität Bern - December 19 2016 Sub-Riemannian geometry Known under many names: Carnot-Carathéodory

### fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

### Solutions to the Hamilton-Jacobi equation as Lagrangian submanifolds

Solutions to the Hamilton-Jacobi equation as Lagrangian submanifolds Matias Dahl January 2004 1 Introduction In this essay we shall study the following problem: Suppose is a smooth -manifold, is a function,

### Brownian Motion and lorentzian manifolds

The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,

### NOTES ON DIFFERENTIAL FORMS. PART 3: TENSORS

NOTES ON DIFFERENTIAL FORMS. PART 3: TENSORS 1. What is a tensor? Let V be a finite-dimensional vector space. 1 It could be R n, it could be the tangent space to a manifold at a point, or it could just

### THE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx

THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two

### DIFFERENTIAL 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

### Topic: 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

### Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples

Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples Vladimir Rubtsov, ITEP,Moscow and LAREMA, Université d Angers Workshop "Geometry and Fluids" Clay Mathematical Institute,

### ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction

ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic

### On the existence of isoperimetric extremals of rotation and the fundamental equations of rotary diffeomorphisms

XV II th International Conference of Geometry, Integrability and Quantization Sveti Konstantin i Elena 2016 On the existence of isoperimetric extremals of rotation and the fundamental equations of rotary

### DIRAC STRUCTURES FROM LIE INTEGRABILITY

International Journal of Geometric Methods in Modern Physics Vol. 9, No. 4 (01) 10005 (7 pages) c World Scientific Publishing Company DOI: 10.114/S0198878100058 DIRAC STRUCTURES FROM LIE INTEGRABILITY

### Warped Products. by Peter Petersen. We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion

Warped Products by Peter Petersen De nitions We shall de ne as few concepts as possible. A tangent vector always has the local coordinate expansion a function the di erential v = dx i (v) df = f dxi We

### Classical 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

### Theorema Egregium, Intrinsic Curvature

Theorema Egregium, Intrinsic Curvature Cartan s second structural equation dω 12 = K θ 1 θ 2, (1) is, as we have seen, a powerful tool for computing curvature. But it also has far reaching theoretical

### Elements of differential geometry

Elements of differential geometry R.Beig (Univ. Vienna) ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014 1. tensor algebra 2. manifolds, vector and covector fields 3. actions under diffeos and

### Holonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012

Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel

### Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups

Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics (Pure and Applied) Rhodes University,

### NEW LIFTS OF SASAKI TYPE OF THE RIEMANNIAN METRICS

NEW LIFTS OF SASAKI TYPE OF THE RIEMANNIAN METRICS Radu Miron Abstract One defines new elliptic and hyperbolic lifts to tangent bundle T M of a Riemann metric g given on the base manifold M. They are homogeneous

### Vortex Equations on Riemannian Surfaces

Vortex Equations on Riemannian Surfaces Amanda Hood, Khoa Nguyen, Joseph Shao Advisor: Chris Woodward Vortex Equations on Riemannian Surfaces p.1/36 Introduction Yang Mills equations: originated from electromagnetism

### EXISTENCE THEORY FOR HARMONIC METRICS

EXISTENCE THEORY FOR HARMONIC METRICS These are the notes of a talk given by the author in Asheville at the workshop Higgs bundles and Harmonic maps in January 2015. It aims to sketch the proof of the

### Chapter 12. Connections on Manifolds

Chapter 12 Connections on Manifolds 12.1 Connections on Manifolds Given a manifold, M, in general, for any two points, p, q 2 M, thereisno natural isomorphismbetween the tangent spaces T p M and T q M.

### CONNECTIONS 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

### Geometric and Spectral Properties of Hypoelliptic Operators

Geometric and Spectral Properties of Hypoelliptic Operators aster Thesis Stine. Berge ay 5, 017 i ii Acknowledgements First of all I want to thank my three supervisors; Erlend Grong, Alexander Vasil ev

### Master Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed

Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like

### PICARD S THEOREM STEFAN FRIEDL

PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A

### Curvature homogeneity of type (1, 3) in pseudo-riemannian manifolds

Curvature homogeneity of type (1, 3) in pseudo-riemannian manifolds Cullen McDonald August, 013 Abstract We construct two new families of pseudo-riemannian manifolds which are curvature homegeneous of

### HOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.

HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### Section 2. Basic formulas and identities in Riemannian geometry

Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l

### Spin(10,1)-metrics with a parallel null spinor and maximal holonomy

Spin(10,1)-metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in