Lecture Notes Topics in Differential Geometry MATH 542

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1 Lecture Notes Topics in Differential Geometry MATH 542 Instructor: Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM August 25, 2014 Author: Ivan Avramidi; File: topicsdiffgeom.tex; Date: December 4, 2014; Time: 13:01

2 Contents 1 Review of Differential Geometry Topological Spaces and Manifolds Tangent Vectors Diffeomorphisms and Flows Dual Space Covectors Differential and Pull-Back Submanifolds Riemannian Metric Tensors Permutation Group Permutation of Tensors Exterior p-forms Exterior Product Orientation of a Vector Space Orientation of a Manifold Volume Form and Hodge Duality Exterior Derivative and Coderivative Pullback of Forms Integration of Fifferential Forms Manifolds with Boundary and Stokes Theorem Lie Derivative Affine Connection and Covariant Derivative Curvature, Torsion and Levi-Civita Connection Parallel Transport Covariant Derivative of Tensors Properties of the Curvature Tensor I

3 II CONTENTS 1.27 Cartan s Structural Equations Poincaré, Integrability, Degree Poincaré Lemma Complex Analysis Degree of a Map Gauss-Bonnet Theorem Laplacian Brouwer Degree Index of a Vector Field Linking Number Groups Groups Group Representations Quotient Spaces Group Actions Free Groups Abelian Groups Groups Finitely Generated and Free Abelian Groups Cyclic Groups Homology Theory Simplicial Homology Groups Simplicial Complexes Simplicial Homology Groups Singular Chains Examples Singular Homology Groups Cycles, Boundaries and Homology Groups Simplicial Homology Betti Numbers and Topological Invariants Some Theorems from Algebraic Topology Examples de Rham Cohomology Groups Hopf Invariants Harmonic Forms topicsdiffgeom.tex; December 4, 2014; 13:01; p. 1

4 CONTENTS III 5 Homotopy Theory Homotopy Groups Fundamental Group Homotopy Type Fundamental Groups of Polyhedra Higher Homotopy Groups Universal Covering Spaces Fibre Bundles Fiber Bundle Vector Bundles Tangent Bundle Cotangent Bundle Tensor Bundles Bundles of Differential Forms Normal Bundle Principal Bundles Associated Bundles Frame Bundle Spin Bundle Trivial Bundles Parallelizable Manifolds Reduction of Bundles Contraction of the Base Space Reduction of the Structure Group G-Structures Connections on Fiber Bundles Connection Lie Groups Connection Connection One-form Horizontal Lift Bibliography 183 Notation 185 topicsdiffgeom.tex; December 4, 2014; 13:01; p. 2

5 IV CONTENTS topicsdiffgeom.tex; December 4, 2014; 13:01; p. 3

6 Chapter 1 Review of Differential Geometry 1.1 Topological Spaces and Manifolds A topology T on a set M is a collection of subsets of M, called open sets, satisfying the conditions 1. T contains the empty set and the whole set M, 2. the intersection of a finite collection of open sets is open, 3. the union of any collection of open sets is open. A topological space (M, T ) is a set M together with a topology T. Any open set containing a point x M is called a neighborhood of x. A sequence (x n ) of points in M is said to converge to a point x, and we write x n x, if every neighborhood of x contains all but a finitely many elements of the sequence, that is, for every neighborhood U of x there is a tail of the sequence contained in U. A map f : M N from a metric space M to a metric space N is said to be continuous at a point x M if for every neighborhood U of the point f (x) N there is a neighborhood V of the point x M such that f (V) U. A map f : M N from a metric space M to a metric space N is said to be continuous if it is continuous at every point. 1

7 2 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY A bijective map f : M N such that both f and f 1 are continuous is called a homeomorphism. A metric on a set M is a function d : M M R satisfying the conditions: for any x, y, z M, d(x, y) 0, d(x, y) = 0, if and only if x = y, d(x, y) = d(y, x), d(x, z) d(x, y) + d(y, z). A metric space (M, d) is a set M together with a metric d. Examples. An open ball of radius ε centered at the point x 0 is the set B ε (x 0 ) = {x M d(x, x 0 ) < ε}. A set U M is called open if for every point x U there is an open ball centered at x and contained in U. Two metrics on a set M are said to be equivalent if they define the same open sets, that is, a subset is open with respect to one metric if and only if it is open with respect to the other metric. Every metric space is a topological space with the natural topology defined as follows: the open sets are the unions of open balls. A set A M is called closed if its complement is open. The limit of a convergent sequence in a closed set belongs to that set. The interior A o of a set A M is the largest open set contained in A. The closure Ā of a set A M is the smallest closed set that contains A. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 4

8 1.1. TOPOLOGICAL SPACES AND MANIFOLDS 3 The boundary of a set A M is the set A = Ā A o. Every subset W of a topological space M is a topological space with the induced topology defined as follows. A set U W is open in W if and only if there is an open set V M such that U = V M. A topological space M is said to be Hausdorff if every two distinct points in M have disjoint neighborhoods. A topological space is discrete if every set containing one point is open. An open cover of a topological space M is a collection of open subsets of M whose union is the whole space M. A topological space is called compact if every open cover of M has a finite subcover. In a compact topological space M every sequence has a convergent subsequence (with a limit in M). For metric spaces the converse is true, that is, if every sequence has a convergent subsequence then the space is compact. A compact subspace W of a topological space M is closed in M. Every closed subspace of a compact topological space is compact. A continuous image of a compact space is compact. Every bijective continuous map f : M N from a compact space M to a space N is a homeomorphism. The Cartesian product M N of topological spaces M and N is a topological space with the product topology defined as follows: the open sets in M N are the unions of the sets of the form U V, where U is an open set in M and V is an open set in N. The product of compact spaces is compact. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 5

9 4 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY A topological space M is called an n-dimensional topological manifold if every point of M has a neighborhood homeomorphic to an open subset of R n. A homeomorphism ϕ : U ϕ(u) from an open subset U of M to an open subset ϕ(u) of R n gives a local coordinate system, or a chart on M. A topological space M is a manifold if it is covered by charts. The collection of all charts {U α } α A is called an atlas. The set ϕ α (U α U β ) is an open subset in R n. The maps f αβ = ϕ α ϕ 1 β : ϕ α (U α U β ) ϕ β (U α U β ) are called the transition maps. An atlas is smooth if all transition maps are smooth. A topological manifold is called a smooth manifold if it is Hausdorff and has a smooth atlas. An atlas determines a smooth structure on the manifold M, which is a collection of all charts that are compatible with the given atlas. Two smooth atlases are said to be compatible if the their union is a smooth atlas. Compatible atlases define the same smooth structure. Let F : M M be a homeomorphism of a topological space M. Then every atlas A = (U α, ϕ α ) on M defines the atlas F(A) = (F(U α ), ϕ α F 1 ). Two atlases A 1 and A 2 on M are said to be equivalent if there is a homeomorphism F : M M such that the atlases A 1 and F(A 2 ) are compatible. Equivalent atlases define equivalent smooth structures. Two smooth structures are equivalent if they are related by a homeomorphism. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 6

10 1.2. TANGENT VECTORS 5 Example. A topological manifold could have inequivalent smooth structures (so called exotic smooth structures). This can only happen for n 4. A topological space M is called an n-dimensional complex manifold if it is Hausdorff and every point of M has a neighborhood homeomorphic to an open subset of C n with holomorphic transition maps. The topological dimension of a n-dimensional complex manifold is 2n. 1.2 Tangent Vectors A tangent vector at a point p 0 M of a manifold M is a map that assigns to each coordinate chart (U α, x α ) about p 0 an ordered n-tuple (Xα, 1..., Xα) n such that n Xβ i = x i β (p 0)Xα j. j=1 x j α Let f : M R be a real-valued function on M. Let p M be a point on M and X be a tangent vector at p. Let (U α, x α ) be a coordinate chart about p. The (directional) derivative of f with respect to X at p (or along X, or in the direction of X) is defined by n ( ) f X p ( f ) = D X ( f ) = (p)xα j. j=1 x j α D X ( f ) does not depend on the local coordinate system. There is a one-to-one correspondence between tangent vectors to M at p and first-order differential operators acting on real-valued functions in a local coordinate chart (U α, x α ) by n X p = Xα j. p j=1 x j α topicsdiffgeom.tex; December 4, 2014; 13:01; p. 7

11 6 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Let M be a manifold and p M be a point in M. The tangent space T p M to M at p is the real vector space of all tangent vectors to M at p. Let (U, x) be a local chart about p. Then the vectors x 1,, p x n form a basis in the tangent space called the coordinate basis, or the coordinate frame. A vector field X on an open set U M in a manifold M is the differentiable assignment of a tangent vector X p to each point p U p X = n j=1 X j (x) x j. Example. 1.3 Diffeomorphisms and Flows Let M be an n-dimensional manifold and N be an m-dimensional manifold and let F : M N be a map from M to N. Let (U α, ϕ α ) α A be an atlas in M and (V β, ψ β ) β B be an atlas in N. We define the maps F αβ = ψ β F ϕ 1 α : ϕ α (U α ) ψ β (V β ) from open sets in R n to R m defined by where a = 1,..., m. y a β = Fa αβ (x1 α,..., x n α), The map F is said to be smooth if Fαβ a are smooth functions of local coordinates xα, i i = 1,..., n. If the map F : M N is bijective and both F and F 1 are differentiable, then F is called a diffeomorphism. This can only happen if m = n. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 8

12 1.4. DUAL SPACE 7 If this is only true in a neighborhood of a point p M, then F is called a local diffeomorphism. Example. A flow on a manifold M corresponding to a vector field X is a one-parameter family of diffeomorphisms such that and for any t, s and ϕ t ϕ s = ϕ t+s, ϕ t : M M ϕ 0 = Id ϕ t = ϕ 1 t d dt ϕ t(x) = X(ϕ t (x)). The corresponding differential operator X( f )(x) = d dt f (ϕ t(x)) = t=0 n j=1 X j (x) f (x) x j is the derivative along the streamline (the integral curve) of the flow through the point p. 1.4 Dual Space Let E be a real n-dimensional vector space. Let {e i } be a basis in E. The set of all linear functionals α : E R on E is called the dual space and denoted by E. Then the linear functionals {σ j } defined by σ j (e i ) = δ j i form a basis in E called the dual basis. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 9

13 8 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Then for any vector v E and any linear functional α E v = n v j e j, j=1 where n α = a j σ j, j=1 v j = σ j (v), a j = α(e j ). 1.5 Covectors Let M be a manifold and p M be a point in M. The space T pm dual to the tangent space T p M at p is called the cotangent space. Let M be a manifold and f : M R be a real valued smooth function on M. Let p M be a point in M. The differential d f T pm of f at p is the linear functional d f : T p R defined by (d f )(v) = v p ( f ). In local coordinates x j the differential is defined by (d f )(v) = n j=1 v j (x) f x j In particular, ( ) (dx i ) = δ i x j j. The differentials {dx i } form a basis for the cotangent space T pm called the coordinate basis. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 10

14 1.6. DIFFERENTIAL AND PULL-BACK 9 Therefore, every linear functional has the form n α = a j dx j. j=1 That is why, the linear functionals are also called differential forms, or 1-forms, or covectors, or covariant vectors. A covector field α is a differentiable assignment of a covector α p T pm to each point p of a manifold n α = a j (x)dx j. j=1 Under a change of local coordinates xα j = xα(x j β ) the differentials transform according to n dxα j xα j = dxβ i. i=1 x i β Therefore, the components of a covector transform as n a α xα j i = a β j. j=1 x i β 1.6 Differential and Pull-Back Let M and N be two manifolds and F : M N be a map from M into N. Let p 0 M be a point in M and X T p0 M be a tangent vector to M at p 0. Let p = p(t), t ( ε, ε), be a curve in M such that Then the differential of F is the map defined by p(0) = p 0, ṗ(0) = X. F : T p0 M T F(p0 )N F X = d dt F(p(t)) t=0. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 11

15 10 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY F does not depend on the curve p(t). Let x i be a local coordinate system in a local chart about p M and y α be a local coordinate system in a local chart about F(p) N and i and α be the coordinate bases for T p M and T F(p) N. The matrix of the linear transformation F in therms of the coordinate bases / y α and / x i is the Jacobian matrix (F ) α i = yα x i, Then the action of the differential F is defined by ( ) m y α F = x j x j y. α α=1 Let X = n i=1 X i i. Then (F X) α = n i=1 y α x i Xi. The pullback is the linear transformation of the cotangent spaces F : T F(p) N T pm taking covectors at F(p) N to covectors at p M, defined as follows. If α T F(p) N, then F α T pm so that F α = α F : T p M R where α : T F(p) N R. That is, for any vector X T p M Diagram. (F α) (X) = α(f X). In local coordinates, [F (dy α )] j = m α=1 y α x j dx j. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 12

16 1.7. SUBMANIFOLDS 11 Let σ = m α=1 σ α dy α. Then F σ = n j=1 m α=1 σ α y α x j dx j, that is, in components, [F σ] j = m α=1 σ α y α x j. Remark. In general, for a map F : M N, the following linear transformations are well defined: the differential F : T p M T F(p) N and the pullback F : T F(p) N T pm. The maps T pm T F(p) N and T F(p)N T p M are not well defined, in general. If dim M = dim N and F : M N is a diffeomorphism, then all these maps are well defined. Explain. 1.7 Submanifolds Implicit description. Let M be an n-dimensional manifold and W M be a subset of M. Then W is an r-dimensional embedded submanifold of M if W is locally described as the common locus of (n r) differentiable independent functions F α (x 1,..., x n ) = 0, α = 1,..., n r, such that the Jacobian matrix has the maximal rank (n r) at each point of the locus, that is, ( ) F α rank x (x) = n r, x W. i topicsdiffgeom.tex; December 4, 2014; 13:01; p. 13

17 12 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY More generally, let M be an n-dimensional manifold and N be an (n r)- dimensional manifold with n > r. Let F : M N be a smooth map. Let q N be a point in N such that the inverse image W = F 1 (q) is nonempty. Suppose that for each point p W the differential F of the map F is surjective, that is, has the maximal rank rank F (p) = n r. Then W is an r-dimensional submanifold of M. The number (n r) is called the codimension of the submanifold W. Explicit description. Let y = (y 1,..., y r ) be local coordinates in a neighborhood of a point q W. Then the submanifold W can be described by a map f : W M (called the inclusion map) such that p = f (q) and x i = f i (y), i = 1,..., n. The r vectors e µ = y µ = n i=1 x i y µ x, i µ = 1,..., r are tangent vector to the submanifold W and form the basis of the tangent space T q W. Note that the differential f has the maximal rank rank f (q) = r. 1.8 Riemannian Metric Let E be a n-dimensional real vector space. The inner product (or scalar product) on E is a symmetric bilinear positivedefinite functional on E E. For a positive-definite inner product the norm of a vector v is defined by v = v, v topicsdiffgeom.tex; December 4, 2014; 13:01; p. 14

18 1.8. RIEMANNIAN METRIC 13 Let {e j } be a basis in E. Then the matrix g i j defined by g i j = e i, e j is a metric tensor, more precisely it gives the components of the metric tensor in that basis. The matrix g i j is symmetric and nondegenerate, that is, g i j = g ji, det g i j 0. For a positive definite inner product, this matrix is positive-definite, that is, it has only positive real eigenvalues. One says, that the metric has the signature (+ +). In special relativity one considers metrics wich are not positive definite but have the signature ( + +). Two vectors v, w E are orthogonal if v, w = 0. A vector u E is called unit vector if The basis is called orthonormal if The inner product is given then by v, w = u = 1. g i j = δ i j. n g i j v i w j. i, j=1 Let v E. Then we can define a linear functional ν E by ν(w) = v, w. Therefore, each vector v E defines a covector ν E called the covariant version of the vector v. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 15

19 14 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Given a basis {e j } in E and the dual basis σ j in E we find ν i = n g i j v j v i = j=1 n g i j v j, j=1 where g i j is the matrix inverse to the matrix g i j. These operations are called lowering the index of a vector and raising the index of a covector. A Riemannian metric on M is a differentiable assignment of a positive definite inner product in each tangent space T p M to the manifold at each point p M. A Riemannian manifold is a pair (M, g) of a manifold with a Riemannian metric on it. Let p M be a point in M and (U, x α ) be a local coordinate system about p. Let i be the coordinate basis in T p M and g α i j = α i, α j be the components of the metric tensor in the coordinate system x α. Let (U β, x β ) be another coordinate system containing p. Then the components of the metric tensor transform as n g α i j = xβ k xβ l g β kl. 1.9 Tensors k,l=1 Let E be a vector space and E be its dual space. Let e i be a basis for E and σ i be the dual basis for E. A tensor of type (p, q) is a multi-linear real-valued functional x i α x j α T : } E {{ E} } E {{ E} R p q The components of the tensor T with respect to the basis e i, σ i are defined by T i 1...i p j 1... j q = T(σ i 1,..., σ i p, e j1,..., e jq ). topicsdiffgeom.tex; December 4, 2014; 13:01; p. 16

20 1.9. TENSORS 15 Then for any covectors n α (a) = α (a) j σ j, where a = 1,..., p, and any vectors j=1 v b = n v k b e k, j=1 where b = 1,..., q, we have T(α (1),..., α (p), v 1,..., v q ) = n n k 1,...,k q =1 j 1,..., j p =1 T j 1... j p k 1...k q α (1) j1 α (p) jp v k1 v k q. The collection of all tensors of type (p, q) forms a vector space denoted by Tq p = } E {{ E} } E {{ E}. p q The dimension of the vector space T p is dim T p q = n p+q. The tensor product of a tensor Q of type (p, q) and a tensor T of type (r, s) is a tensor Q T of type (p+r, q+s) defined by: α 1,..., α p, β 1,..., β r E, v 1,..., v q, w 1,..., w s E (Q T)(α 1,..., α p, β 1,..., β r, v 1,..., v q, w 1,..., w s ) = Q(α 1,..., α p, v 1,..., v q )T(β 1,..., β r, w 1,..., w s ). The components of the tensor product Q T are (Q T) i 1...i p j 1... j r k 1...k q l 1...l s = Q i 1...i p k 1...k q T j 1... j r l 1...l s. Thus, : T p q T r s T p+r q+s. The tensor product is associative. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 17

21 16 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY The basis in the space Tq p is e i1 e ip σ j 1 σ j q, where 1 i 1,..., i p, j 1,..., j q n. A tensor T of type (p, q) has the form T = n n j 1,..., j q =1 i 1,...,i p =1 T i 1...i p j 1... j q e i1 e ip σ j 1 σ j q. Let p, q 1 and 1 r p, 1 s q. The (r, s)-contraction of tensors of type (p, q) is the map tr r s : T p q T p 1 q 1 defined by (tr r s T) i 1...i p 1 j 1... j q 1 = n k=1 T i 1...i r 1 ki r...i p 1 j 1... j s 1 k j s... j q. A tensor field on a manifold M is a smooth assignment of a tensor at each point of M. Let x i α = x i α(x β ) be a local diffeomorphism. Let T be a tensor of type (p, q). Then T (α) i 1...i p j 1... j q (x α ) = n n k 1,...,k p =1 l 1,...,l q =1 x i 1 α x k 1 β xi p α x l 1 β x k p β j 1 β x x l q α x j q α T (β) k 1...k p l 1...l q (x β ) Einstein Summation Convention In any expression there are two types of indices: free indices and repeated indices. Free indices appear only once in an expression; they are assumed to take all possible values from 1 to n. The position of all free indices in all terms in an equation must be the same. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 18

22 1.10. PERMUTATION GROUP 17 Repeated indices appear twice in an expression. It is assumed that there is a summation over each repeated pair of indices from 1 to n. The summation over a pair of repeated indices in an expression is called the contraction. Repeated indices are dummy indices: they can be replaced by any other letter (not already used in the expression) without changing the meaning of the expression. Indices cannot be repeated on the same level. That is, in a pair of repeated indices one index is in upper position and another is in the lower position. There cannot be indices occuring three or more times in any expression Permutation Group A group is a set G with an associative binary operation, : G G G with identity, called the multiplication, such that each element has an inverse. A transformation of a set X is a bijective map g : X X. The set of all transformations of a set X forms a group Aut(X), with composition of maps as group multiplication. Any subgroup of Aut(X) is a transformation group of the set X. The transformations of a finite set X are called permutations. The group S p of permutations of the set Z n = {1,..., p} is called the symmetric group of order p. The order of the symmetric group S p is S p = p!. Any subgroup of S p is called a permutation group. An elementary permutation is a permutation that exchanges the order of only two elements. Every permutation can be realized as a product of elementary permutations. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 19

23 18 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY A permutation that can be realized by an even number of elementary permutations is called an even permutation. A permutation that can be realized by an odd number of elementary permutations is called an odd permutation. The parity of a permutation does not depend on the representation of a permutation by a product of the elementary ones. The sign of a permutation ϕ S p, denoted by sign(ϕ) (or simply ( 1) ϕ ), is defined by { +1, if ϕ is even, sign(ϕ) = ( 1) ϕ = 1, if ϕ is odd 1.11 Permutation of Tensors Let S p be the symmetric group of order p. Then every permutation ϕ S p defines a map ϕ : T p T p, which assigns to every tensor T of type (0, p) a new tensor ϕ(t), called a permutation of the tensor T, of type (0, p) by: v 1,..., v p ϕ(t)(v 1,..., v p ) = T(v ϕ(1),..., v ϕ(p) ). Let (i 1,..., i p ) be a p-tuple of integers. Then a permutation ϕ : Z p Z p defines an action ϕ(i 1,..., i p ) = (i ϕ(1),..., i ϕ(p) ). The components of the tensor ϕ(t) are obtained by the action of the permutation ϕ on the indices of the tensor T ϕ(t) i1...i p = T iϕ(1)...i ϕ(p). The symmetrization of the tensor T of the type (0, p) is defined by Sym(T) = 1 ϕ(t). p! ϕ S p topicsdiffgeom.tex; December 4, 2014; 13:01; p. 20

24 1.11. PERMUTATION OF TENSORS 19 The symmetrization is also denoted by parenthesis. The components of the symmetrized tensor Sym(T) are given by T (i1...i p ) = 1 T iϕ(1)...i p! ϕ(p). ϕ S p The anti-symmetrization of the tensor T of the type (0, p) is defined by Alt(T) = 1 sign (ϕ)ϕ(t). p! ϕ S p The anti-symmetrization is also denoted by square brackets. The components of the anti-symmetrized tensor Alt(T) are given by Examples. T [i1...i p ] = 1 sign (ϕ)t iϕ(1)...i p! ϕ(p). ϕ S p A tensor T of type (0, p) is called symmetric if for any permutation ϕ S p ϕ(t) = T. A tensor T of type (0, p) is called anti-symmetric if for any permutation ϕ S p ϕ(t) = sign (ϕ)t. An anti-symmetric tensor of type (0, p) is called a p-form. Let (i 1,..., i n ) be an n-tuple of integers 1 i 1,..., i n n. The completely anti-symmetric (alternating) Levi-Civita symbols are defined by 1 if (i ε i 1...i n 1,..., i n ) is an even permutation of (1,..., n) = ε i1...i n = 1 if (i 1,..., i n ) is an odd permutation of (1,..., n) 0 otherwise The product of Levi-Civita symbols is equal to ε i 1...i n ε j1... j n = n!δ i 1 [ j 1 δ i n j n ]. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 21

25 20 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY The contraction of this identity over k indices gives ε i 1...i n k m 1...m k ε j1... j n k m 1...m k = k!(n k)!δ i 1 [ j 1 δ i n k j n k ]. In particular, ε m 1...m n ε m1...m n = n!. The determinant of a n n matrix A = (A i j) is defined by det A = sign (ϕ)a 1 ϕ(1) A n ϕ(n), ϕ S n It can be written as det A = 1 n! εi 1...i n ε j1... j n A j 1 i1... A j n in Exterior p-forms An exterior p-form (or simply a p-form) is an anti-symmetric covariant tensor α T p of type (0, p). The collection of all p-forms forms a vector space Λ p, which is a vector subspace of T p Λ p T p. In particular, Λ 0 = R and Λ 1 = T 1 = E. In other words, a 0-form is a smooth function, and a 1-form is a covector field. Let α Λ p be a p-form. Let e i be a basis in E and σ i be the dual basis in E. The components of the p-form α are α i1...i p = α(e i1,... e ip ). The components are completely anti-symmetric in all indices. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 22

26 1.12. EXTERIOR P-FORMS 21 In particular, under a permutation of any two indices the form changes sign α... i... j... = α... j... i..., which means that the components vanish if any two indices are equal α... i... i... = 0 (no summation!). Thus, all non-vanishing components have different indices. Therefore, the values of all components α i1...i p are completely determined by the values of the components with the indices i 1,..., i p reordered in strictly increasing order 1 i 1 < < i p n. Therefore, the dimension of the space Λ p is equal to the number of distinct increasing p-tuples of integers from 1 to n. dim Λ p = ( ) n = p n! p!(n p)!. In particular, dim Λ 0 = dim Λ n = 1, dim Λ 1 = dim Λ n 1 = n, etc. There are no p-forms with p > n. Similarly to the norm of vectors and covectors we define the inner product of exterior p-forms α and β in a Riemannian manifold by (α, β) = 1 p! gi 1 j1 g i 1 j p α i1...i p β j1... j p. This enables one to define also the norm of an exterior p-form α by α = (α, α) topicsdiffgeom.tex; December 4, 2014; 13:01; p. 23

27 22 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY 1.13 Exterior Product If α is an p-form and β is an q-form then the exterior (or wedge) product. of α and β is an (p + q)-form α β defined by α β = (p + q)! Alt (α β). p!q! In components (α β) i1...i p+q = (p + q)! α [i1...i p!q! p β ip+1...i p+q ]. The exterior product has the following properties (α β) γ = α (β γ) (associativity) α β = ( 1) deg(α)deg(β) β α (anticommutativity) (α + β) γ = α γ + β γ (distributivity). The exterior square of any p-form α of odd degree p (in particular, for any 1-form) vanishes α α = 0. The exterior algebra Λ (or Grassmann algebra) is the set of all forms of all degrees, that is, Λ = Λ 0 Λ n. The dimension of the exterior algebra is dim Λ = n p=0 ( ) n = 2 n. p A basis of the space Λ p is σ i 1 σ i p, (1 i 1 < < i p n). topicsdiffgeom.tex; December 4, 2014; 13:01; p. 24

28 1.13. EXTERIOR PRODUCT 23 A p-form α can be represented in one of the following ways α = 1 p! α i 1...i p σ i 1 σ i p = α i1...i p σ i1 σ ip. i 1 < <i p The exterior product of a p-form α and a q-form β can be represented as α β = 1 p!q! α [i 1...i p β ip+1...i p+q ]σ i 1 σ i p+q. A collections of 1-forms α 1,..., α p Λ 1 is linearly dependent if and only if α 1 α p = 0. More generally, let σ 1,..., σ n Λ 1 be a collection of 1-forms and α j = n A j iσ i, 1 j n, i=1 Then α 1 α n = (det A i j) σ 1 σ n. This means that for a local diffeomorphism x i α = x i α(x β ), i = 1,..., n, there holds dx 1 α dx n α = J αβ (x β )dx 1 β dx n β. where J αβ (x β ) = det x i α x j β is the Jacobian. The interior product of a vector v and a p-form α is a (p 1)-form i v α defined by, for any v 1,..., v p 1, i v α(v 1,..., v p 1 ) = α(v, v 1,..., v p 1 ). topicsdiffgeom.tex; December 4, 2014; 13:01; p. 25

29 24 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY In particular, if p = 1, then i v α is a scalar i v α = α(v) and if p = 0, then by definition i v α = 0. In components, (i v α) i1...i p 1 = v j α ji1...i p 1. For any α Λ p, β Λ q, i v (α β) = (i v α) β + ( 1) p α i v β Orientation of a Vector Space Let E be a vector space. Let {e i } = {e 1,..., e n } and {e j } = {e 1,..., e n} be two different bases in E related by e i = Λ j ie j, where Λ = (Λ i j) is a transformation matrix. Note that the transformation matrix is non-degenerate det Λ 0. Since the transformation matrix Λ is invertible, then the determinant det Λ is either positive or negative. If det Λ > 0 then we say that the bases {e i } and {e i } have the same orientation, and if det Λ < 0 then we say that the bases {e i } and {e i } have the opposite orientation. If the basis {e i } is continuously deformed into the basis {e j }, then both bases have the same orientation. Since det I = 1 > 0 and the function det : GL(n, R) R is continuous, then a one-parameter continuous transformation matrix Λ(t) such that Λ(0) = I preserves the orientation. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 26

30 1.15. ORIENTATION OF A MANIFOLD 25 This defines an equivalence relation on the set of all bases on E called the orientation of the vector space E. This equivalence relation divides the set of all bases in two equivalence classes, called the positively oriented and negatively oriented bases. A vector space together with a choice of what equivalence class is positively oriented is called an oriented vector space Orientation of a Manifold Let M be a manifold and let T p M be the tangent space at a point p M. Let (U, x) be a local coordinate patch about a point p M. Then the vectors form a basis in T p M. x i, i = 1,..., n Let (U, x ) be another local coordinate system about a point p, that is, there is a local diffeomorphism x i = x i (x ). Then the vectors form another basis in T p M. = x j x i x i x j The orientation of the bases { i } and { j } is the same (or consistent) if ( ) x i det > 0. x j If it is possible to choose an orientation of all tangent spaces T p M at all points in a continuous fashion, then the orientation of all tangent spaces is consistent. A manifold M is called orientable if there is an atlas such that the orientation of all charts of this atlas can be chosen consistently, that is, the Jacobians of all transition functions have positive determinant. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 27

31 26 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Each connected orientable manifold has exactly two possible orientations. One orientation can be declared positive, then the other orientation is negative. An orientable manifold with a chosen orientation is called oriented. Remarks. If a manifold can be covered by a single coordinate chart then it is orientable. Not all manifolds are orientable. Transport of the orientation. Let p, q M be two points in a manifold M and C(t) be a curve in M connecting p and q, i.e. C(0) = p, C(1) = q. Let {e i (t)} be a basis in T C(t) M that continuously depends on t [0, 1]. Then the orientation of the basis e i (1) is uniquely determined by the orientation of the basis e i (0). Thus, the orientation is transported along a curve in a unique way. Note that the transportation of the basis is not unique, in general. Only the transportation of the orientation is! Given a point p and another point q, the orientation at the point q does, in general, depend on the curve C(t) connecting the points p and q. If a manifold is orientable, then the transportation of the orientation from one point to another does not depend on the curve connecting the points. If there exists a closed curve C(t) in M such that the transport of the orientation along C leads to a reversal of orientation, then M is nonorientable. Example. Möbius Band. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 28

32 1.16. VOLUME FORM AND HODGE DUALITY Volume Form and Hodge Duality Let e i be a basis in a vector space E, σ i be the dual basis of 1-forms in E, g = (g i j ) be a Riemannian metric. The n-form where vol = g σ 1 σ n g = det g i j is called the Riemannian volume element (or volume form). The components of the volume form are vol (e i1,..., e in ) = g ε i1...i n = E i1...i n. The volume form allows one to define the duality of p-forms and (n p)- vectors. For each p-form A i1...i p one assigns the dual (n p)-vector by à j 1... j n p = 1 p! E j 1... j n p i 1...i p A i1...i p. Similarly, for each p-vector A i 1...i p one assigns the dual (n p)-form by à j1... j n p = 1 p! E j 1... j n p i 1...i p A i 1...i p. By lowering and raising the indices of the dual forms we can define the duality of forms and poly-vectors separately. The Hodge star operator : Λ p Λ n p maps any p-form α to a (n p)-form α dual to α defined as follows. For each p-form α the form α is the unique (n p)-form such that α α = (α, α)vol. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 29

33 28 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY In particular, 1 = vol, vol = 1. In components, this means that ( α) ip+1... i n = 1 p! ε i 1... i p i p+1...i n g g i 1 j1 g i p j p α j1... j p = 1 p! 1 g g ip+1 j p+1 g in j n ε j 1... j p j p+1... j n α j1... j p. For any p-form α there holds 2 α = ( 1) p(n p) α. In particular, if n is odd, then for any p 2 = Id. Let α be a 1-form and v be the corresponding vector, that is, v i = g i j α j. Then α = i v vol. A collection {ω (1),..., ω (n 1) } of (n 1) 1-forms defines a 1-form α by α = [ ω (1) ω (n 1) ]. Then and In components, α = ( 1) n 1 ω (1) ω (n 1) ω (1) ω (n 1) α = (α, α)vol. α j = g jk E i 1...i n 1 k ω i1 (1) ω in 1 (n 1). If the 1-forms {ω (1),..., ω (n 1) } are linearly dependent, then α = 0. If the collection of 1-forms {ω (1),..., ω (n 1) } is linearly independent, then {ω (1),..., ω (n 1), α} are linearly independent and form a basis in E. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 30

34 1.17. EXTERIOR DERIVATIVE AND CODERIVATIVE 29 Similarly, a collection {v (1),..., v (n 1) } of (n 1) vectors defines a covector α by: for any vector v or, in components, and the corresponding vector α(v) = vol (v (1),..., v (n 1), v) α j = g ε i1...i n 1 jv i 1 (1) v i n 1 (n 1). N i = g ik g ε i1...i n 1 kv i 1 (1) v i n 1 (n 1) The vector N is orthogonal to all vectors {v (1),..., v (n 1) }, that is, (N, v ( j) ) = g ik N i v k ( j) = 0, ( j = 1,..., n 1). The n-tuple {v 1,..., v n 1, N} forms a positively oriented basis Exterior Derivative and Coderivative The exterior derivative is a linear map d : Λ p Λ p+1 defined as follows. Let α be a p-form α = 1 p! α i 1...i p dx i 1 dx i p. The exterior derivative of α is a (p + 1)-form dα defined by dα = 1 p! i 1 α i2...i p+1 dx i 1 dx i 2 dx i p+1. In components (dα) i1 i 2...i p+1 = (p + 1) [i1 α i2...i p+1 ] p+1 = ( 1) k 1 ik α i1...i k 1 i k+1...i p+1 k=1 topicsdiffgeom.tex; December 4, 2014; 13:01; p. 31

35 30 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY For any p-form d 2 = 0. For any p-form α Λ p and any q-form β Λ q there holds d(α β) = (dα) β + ( 1) p α (dβ). The coderivative is a linear map δ : Λ p Λ p 1 defined by δ = 1 d = ( 1) (n p+1)(p 1) d The coderivative of a p-form α is the (p 1)-form δα define by (δα) i1...i p 1 = g i1 j 1... g ip 1 j p 1 1 g j ( g g jk g j 1k1 g j p 1k p 1 α kk1...k p 1 ). For any p-form δ 2 = 0. For a 1-form α, δα is a 0-form δα = 1 g i ( g g i j α j ). The Hodge Laplacian is a linear map : Λ p Λ p defined by = dδ + δd. It is easy to see that Laplacian commutes with both d and δ. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 32

36 1.18. PULLBACK OF FORMS Pullback of Forms Let M be a n-dimensional manifold and W be a r-dimensional manifold. Let F : M N be a smooth map of a manifold M to a manifold N. Let p M be a point in M and q = F(p) N be the image of p in N. Let x i, (i = 1,..., n), be a local coordinate system about p and y µ, (µ = 1,..., r), be a local coordinate system about q so that The pullback of p-forms is the map defined as follows. y µ = y µ (x). F : Λ p N Λ p M Let α Λ p N be a p-form on N. The pullback of α is a p-form F α on M defined by: for any vectors v 1,..., v p In local coordinates In components (F α)(v 1,..., v p ) = α(f v 1,..., F v p ). F α = 1 p! α µ 1...µ p (y(x))dy µ 1 dy µ p = 1 y µ 1 p! x yµp i 1 x α i µ p 1...µ p (y(x))dx i 1 dx i p (F α) i1...i p = yµ 1 x i 1 yµp x i p α µ 1...µ p (y(x)) For any two forms α and β F (α β) = (F α) (F β) F commutes with exterior derivative, that is, for any p-form α F (dα) = d(f α). topicsdiffgeom.tex; December 4, 2014; 13:01; p. 33

37 32 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY 1.19 Integration of Fifferential Forms Let M be an n-dimensional manifold with local coordinates x i, i = 1,..., n. Let 0 p n and U be an oriented region in R p with orientation o and coordinates u µ, µ = 1,..., p. Let F : U M be a smooth map given locally by x i = F i (u). Then the image F(U) M of the set U is called a p-subset of M and the collection (U, o, F) is called an oriented parametrized p-subset of M. Usually, the differential F has rank p (that is, F(U) is a submanifold) almost everywhere. Let α Λ p be a p-form on M α = 1 p! α i 1...i p dx i 1 dx i p. The integral of α over an oriented parametrized p-subset F(U) is defined by α = F α. F(U) In more detail, ( ) α = o(u) (F α) F(U) U u,..., du 1 du p 1 u p ( ) = o(u) α F U u,..., F 1 du 1 du p u p = 1 p! o(u) α i1...i p (x(u)) xi 1 u xip 1 u p du1 du p. U The integral of the form α over U reverses sign if the orientation of U is reversed. The integral is independent of the parametrization of a p-subset. U topicsdiffgeom.tex; December 4, 2014; 13:01; p. 34

38 1.20. MANIFOLDS WITH BOUNDARY AND STOKES THEOREM 33 Let M be an n-dimensional manifold and W be an r-dimensional manifold. Let ϕ : M W be a smooth map. Let U R p be an oriented region in R n and F : U M be an oriented parametrized p-subset of M. Then ψ = ϕ F : U W is an oriented parametrized p-subset of W. Let α Λ p W be a p-form on W. Then α = ψ α = (F ϕ )α = F (ϕ α) = ϕ α ψ(u) U U U F(U) Let S = F(U) be an oriented subset of M. Then ψ(u) = ϕ(f(u)) = ϕ(s ) is an oriented subset of W. The general pullback formula takes the form α = ϕ α. ϕ(s ) 1.20 Manifolds with Boundary and Stokes Theorem Recall that an open ball in R n is the set B ɛ (x 0 ) = {x R n x x 0 < ε} S Let us consider also sets H ɛ (x 0 ) = {x R n x x 0 < ε, x n x n 0 0}. Such sets are called half-open balls. A n-dimensional manifold with boundary consists of the the interior M o and the boundary M. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 35

39 34 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY The interior M o is a genuine n-dimensional manifold such that all its points have neighborhoods diffeomorphic to open balls in R n. The boundary M is a subset of M such that all its points have neighborhoods diffeomorphic to half-open balls. Usually, the boundary M is itself an (n 1)-dimensional submanifold of M (without boundary). Boundary may be disconnected. It can also be not smooth. Local coordinates (x 1,..., x n 1, x n ) in M in the neighborhoods of points on the boundary can always be chosen in such a way that (x 1,..., x n 1 ) are the coordinates along the boundary and 0 x n < δ with some δ. A compact manifold is a manifold which is closed and bounded (say, as a submanifold of some R N ). A closed manifold is a manifold which is compact and does not have a boundary. Let M be an n-dimensional orientable manifold with boundary M, which is an (n 1)-dimensional manifold without boundary. Let M be oriented. Then an orientation on M naturally induces an orientation on M. Let p M and {e 2,..., e n } be a basis in T p M. Let N T p M be a tangent vector at p that is transverse to M and points out of M. Then {N, e 2,..., e n } forms a basis in T p M. Then, by definition, the basis {e 2,..., e n } has the same orientation as the basis {N, e 2,..., e n }. That is, {e 2,..., e n } is positively oriented in M if {N, e 2,..., e n } is positively oriented in M. Stokes Theorem. Let M be an n-dimensional manifold and V be a p- dimensional compact oriented submanifold with boundary V in M. Let ω Λ p 1 M be a smooth (p 1)-form in M. Then dω = ω. V V topicsdiffgeom.tex; December 4, 2014; 13:01; p. 36

40 1.21. LIE DERIVATIVE Lie Derivative Let X be a vector field on a manifold M. The Lie derivative of a tensor of type (p, q) along the vector X is a linear map L X : T p q M T p q M defines as follows. Let ϕ t : M M be the flow generated by X. Since the flow ϕ t : M M is a diffeomorphism it naturally acts on general tensors of type (p, q), that is, ϕ t : (T p q ) ϕt (x)m (T p q ) x M. Namely, ϕ t T is a tensor field of type (p, q) defined by (ϕ t T) k 1...k p i 1...i q (x) = ϕ j1 t (x) x i 1 ϕ j q t (x) x k 1 x i q ϕ m 1 t (x) x kp ϕ m p t (x) T m1...mp j 1... j q (ϕ t (x)) Let T be a tensor field of type (p, q) on M. The Lie derivative of T with respect to X is a tensor field L X T of type (p, q) defined by (L X T) x = d ( ϕ dt t T ) x. t=0 The Lie derivative of a tensor field T of type (p, q) with respect to a vector field X is given in local coordinates by (L X T) k 1...k p i 1...i q = X j j T k 1...k p i 1...i q + T k 1...k p ji 2...i q i1 X j + + T k 1...k p i 1...i q 1 j i q X j T jk 2...k p i 1 i 2...i q j X k 1 T k 1...k p 1 j j X k p The Lie derivative of the function f along the vector field X is the rate of change of f along the flow generated by X, L X f = X( f ). i 1...i q topicsdiffgeom.tex; December 4, 2014; 13:01; p. 37

41 36 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY The Lie bracket of two vector fields X and Y is a vector field [X, Y] such that for any smooth function f on M [X, Y]( f ) = X(Y( f )) Y(X( f )). Notice that [X, Y] = [Y, X]. In local coordinates the Lie bracket is given by [X, Y] i = X j j Y i Y j j X i. The Lie derivative of the vector field Y along the vector field X is given by L X Y = [X, Y]. Let M be a manifold and W be a submanifold of M. Let X and Y be vector fields on M tangent to W. Then the Lie bracket [X, Y] is also tangent to W. The Lie derivative of a p-form α with respect to a vector field X is given by (L X α) i1...i p = X j j α i1...i p + α ji2...i p i1 X j + + α i1...i p 1 j ip X j In particular, for a 1-form α we have (L X α) i = X j j α i + α j i X j. In particular, for a tensor g i j of type (0, 2) we obtain (L X g) i j = X k k g i j + g ik j X k + g k j i X k. For any two tensors T and R and a vector field X the Leibnitz rule holds L X (T R) = (L X T) R + T (L X R) Let α be a p-form, β be a q-form and X be a vector field on M. Then the Leibnitz rule holds L X (α β) = (L X α) β + α (L X β) topicsdiffgeom.tex; December 4, 2014; 13:01; p. 38

42 1.22. AFFINE CONNECTION AND COVARIANT DERIVATIVE 37 The Lie derivative commutes with the exterior derivative, that is, Cartan Formula. L X d = dl X. L X = i X d + di X Let g i j be a Riemannian metric on an n-dimensional manifold M and vol = g dx 1 dx n be the Riemannian volume form. Let X be a vector field on M. Then L X vol = d(i X vol ) = ( div X)vol, where div X is a scalar function defined by div X = L X vol = 1 g i ( g X i ). The scalar div X is called the divergence of the vector field X Affine Connection and Covariant Derivative Let M be an n-dimensional manifold. An affine connection is an operator : C (T M) C (T M) C (T M) that assigns to two vector fields X and Y a new vector field X Y, that is linear in both variables, that is, for any a, b R and any vector fields X, Y and Z, X (ay + bz) = a X Y + b X Z ax+by Z = a X Z + b Y Z and satisfies the Leibnitz rule, that is, for any smooth function f C (M) and any two vector fields X and Y, X ( f Y) = X( f )Y + f X Y = (d f )(X)Y + f X Y topicsdiffgeom.tex; December 4, 2014; 13:01; p. 39

43 38 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Let x µ, µ = 1,..., n, be local coordinates and µ be the basis of vector fields. It will be called a coordinate frame for the tangent bundle. A basis of vector fields e i = e µ i µ, i = 1,..., n, for the tangent bundle T M is called a frame. Let σ j = σ j µdx µ be the dual frame of 1-forms. Then and σ i (e j ) = σ i µe µ j = δ i j σ i µe ν i = δ ν µ. We will denote the action of frame vector fields on functions by e i ( f ) = e µ i µ = f i. For any frame the commutator of the frame vector fields defines the commutation coefficients [e i, e j ] = C k i je k. In components, e ν j i eν i j = eµ i µe ν j eµ j µe ν i = C k i je ν k. That is, C k i j = σ k ([e i, e j ]) = σ k ν ( e µ i µe ν j eµ j µe ν i ). The symbols ω i jk defined by ω i k j = σ i ( e j e k ) are called the coefficients of the affine connection. We denote Then i = ei. i e j = ω k jie k, topicsdiffgeom.tex; December 4, 2014; 13:01; p. 40

44 1.23. CURVATURE, TORSION AND LEVI-CIVITA CONNECTION 39 Then, if X = X i e i is a vector field, then X = X i i. If Y = Y j e j is another vector field then X Y = X i i Y k e k where i Y k = Y k i + ω k jiy j The tensor field Y of type (1, 1) with components i Y k is called the covariant derivative of the vector field Y. In the coordinate frame e i = i the covariant derivative takes the form i Y k = i Y k + ω k jiy j Curvature, Torsion and Levi-Civita Connection Let X and Y be vector fields on a manifold M. Then the vector field T (X, Y) = X Y Y X [X, Y] defines a tensor field T of type (1, 2), called the torsion, so that for any 1-form σ T(σ, X, Y) = σ(t (X, Y)). The affine connection is called torsion-free (or symmetric) if the torsion vanishes, that is, for any X, Y, X Y Y X = [X, Y] The components of the torsion tensor are defined by T i jk = σ i (T (e j, e k )). In the coordinate frame the components of the torsion tensor are given by T i jk = ω i k j ω i jk. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 41

45 40 CHAPTER 1. REVIEW OF DIFFERENTIAL GEOMETRY Let X, Y and Z be vector fields on a manifold M. Then the vector field R(X, Y)Z = { [ X, Y ] [X,Y] } Z defines a tensor field R of type (1, 3), called the Riemann curvature, so that for any 1-form σ R(σ, Z, X, Y) = σ(r(x, Y)Z). The affine connection is called flat if the curvature vanishes, that is, for any X, Y, Z, [ X, Y ]Z = [X,Y] Z. The components of the curvature tensor are defined by R i jkl = σ i (R(e k, e l )e j )). The components of the curvature tensor have the form R i jkl = ω i jl k ω i jk l + ω i mkω m jl ω i mlω m jk C m klω i jm. In the coordinate frame the components of the curvature tensor are given by R i jkl = k ω i jl l ω i jk + ω i mkω m jl ω i mlω m jk. For a Riemannian manifold (M, g) the metric tensor g has the components g i j = g(e i, e j ) = g µν e µ i eν j. This metric is used to lower and raise the frame indices. Let (M, g) be a Riemannian manifold and be an affine connection on M. Then the connection is called compatible with the metric g if for any vector fields X, Y and Z it satisfies the condition Z(g(X, Y)) = g( Z X, Y) + g(x, Z Y). An affine connection that is torsion-free and compatible with the metric is called the Levi-Civita connection. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 42

46 1.24. PARALLEL TRANSPORT 41 Each Riemannian manifold has a unique Levi-Civita connection. We define ω i jk = g im ω m jk C i jk = g im C m jk. The coefficients of the Levi-Civita connection are given by ω i jk = 1 2 ( gi j k + g ik j g jk i + C ki j + C jik C i jk ). The coefficients of the Levi-Civita connection in a coordinate frame are called Christoffel symbols and denoted by Γ i jk Γ i jk = 1 2 gim ( j g mk + k g jm m g jk ). Christoffel symbols have the following symmetry property Γ i jk = Γ i k j. The coefficients of the Levi-Civita connection in an orthonormal frame have the form ω i jk = 1 ( ) Cki j + C jik C i jk. 2 They have the following symmetry properties ω i jk = ω jik. In a coordinate basis for a torsion-free connection we have the following identities (called the Ricci identities): [ i, j ]Y k = R k li jy l 1.24 Parallel Transport Let x 0 and x 1 be two points on a manifold M and C be a smooth curve connecting these points described locally by x i = x i (t), where t [0, 1] and x(0) = x 0 and x(1) = x 1. The tangent vector to C is defined by X = ẋ(t), where the dot denotes the derivative with respect to t. topicsdiffgeom.tex; December 4, 2014; 13:01; p. 43