The Algebra of Tensors; Tensors on a Vector Space Definition. Suppose V 1,,V k and W are vector spaces. A map F : V 1 V k is said to be multilinear if it is linear as a function of each variable seperately: F (v 1,,av i + a v i,,v k)=af (v 1,,v i,,v k )+a F (v 1,,v i,,v k). (A multilinear function of two variables is generally called bilinear.) Examples. (1) The dot product in R n is a scalar-valued bilinear function of two vectors, used to compute lenths of vectors and angles between them. (2) The cross product in R 3 is a vector-valued bilinear function of two vectors, used to compute areas of paralleograms and to find a third vector orthogonal to the given ones. (3) The determinant is a real-valued multilinear function of n vectors in R n, used to detect linear independence and to compute the volume of the parallelepiped spanned by the vectors. Let V be a finite-dimensional vector space. Definition. (1) A covector is a real-valued linear functional on V. (2) The dual space V of V is the space of covectors on V. (3) Denote the natural pairing V V R by either of the notations (w, X) w, X or (w, X) w(x), w V,X V. Definition. (1) A covariant k-tensor on V is a multilinear map F : V V R. }{{} k copies (2) A contravariant l-tensor on V is a multilinear map F : V V }{{ R. } l copies (3) A tensor of type ( k, also called k-covariant, l-contravariant tensor is a multilinear map F : V V }{{ } l copies V V R. }{{} k copies Typeset by AMS-TEX 1
2 Definition. (1) The space of all covariant k-tensors on V is denoted by T k (V ). (2) The space of all contravariant l-tensors on V is denoted by T l (V ). (3) The space of all mixed ( k -tensors on V is denoted by T k l (V ). T k 0 (V )=T k (V ), T 0 l (V )=T l(v ), T 1 (V )=V, T 1 (V )=V = V, T 0 (V )=R. Examples (Covariant Tensors) (1) Every linear map ω : V R is multilinear, so a covariant 1-tensor is just a covector. Thus T 1 (V )=V. (2) A covariant 2-tensor on V is a real-valued bilinear function of two vectors, also called a bilinear form. For example, any inner product on V is a covariant 2-tensor. (3) The determinant, thought of as a function of n vectors, is a covariant n- tensor on R n. (4) Suppose ω, η V. Define a map ω η : V V R by ω η(x, Y )=ω(x)η(y ), where the product on the right is ordinary multiplication of real numbers. The linearity of ω and η guarantees that ω η is a bilinear function of X and Y, i.e. a covaiant 2-tensor. The last example can be generalized to tensors of any rank as follows. Definition. It F Tl k(v ) and G = T q p by k+p (V ), the tensor F G Tl+q (V ) is defined F G(ω 1,,ω l+q,x 1,,X k+p ) = F (ω 1,,ω l,x 1, X k )G(ω l+1,,ω l+q,x k+1,,x k+p ). Proposition 1. Let V be a real vector space of dimension n, let (E i ) be any basis for V, and let (ε i ) be the dual basis, defined by ε i (E j )=δj i. A basis for T l k (V ) is given by the set of all tensors of the form, as the indices i p, j q ranges from 1 to n. The space Tl k (V ) therefore has dimension n k+l. Proof. These tensors act on basis elements by (ε s1,,ε s l,e r1,,e rk )=δ s1 j 1 δ s l j l δ i1 r 1 δ i k rk. Let B denote the set {, 1 i 1,,i k n, 1 j 1,,j l n}.
(I) Claim: B spans Tl k(v ). Any tensor F T l k (V ) can be written in terms of this basis as F = F j1 j l, where (11.1) F j1 j l = F (ε j1,,ε j l,e i1,,e ik ). Indeed, we heve F j1 j l (ε s1,,ε s l,e r1,,e rk ) =F j1 j l δ s1 j 1 δ s l j l δr i1 1 δ i k rk =F s1,,s k r 1,,r l =F (ε s1,,ε s l,e r1,,e rk ). By multilinearity, a tensor is determined by its action on sequences of basis vectors, so this proves the claim. (II) Claim: B is independent. Suppose some linear combination equals zero: F j1 j l =0. Apply this to any sequence (ε s1,,ε s l,e r1,,e rk ) of basis vectors. By the same computation as above, this implies that each coefficient F j1 j l is zero. The proof shows, by the way, that the components F j1 j l of a tensor F in terms of the basis ternsor in B are given by (11.1). It is useful to see explicitly what this proposition means for tensors of low rank. (1) k =0: T 0 (V )=R, so dim T 0 (V )=1=n 0. (2) k =1: T 1 (V )=V has dimension n = n 1. (3) k =2: T 2 (V ) is the space of bilinear forms on V. Any bilinear foem can be written uniquely as T = T ij ε i ε j, where (T ij ) is an arbitrary n n matrix. Thus dim T 2 (V )=n 2. 3
4 Tensor Bundles and Tensor Fields A ( k -tensor at p M is just an element of T k l (T p M). Definition. The budle of ( k -tensors on M is Tl k M = Tl k (T pm), p M where denotes the disjoint union. To see that each of these tensor bundles is a vector bundle, define the projection π : Tl km M to be the map that simply sends F T l k(t pm) top. T 0 M = T 0 M = M R, T 1 M = T M, T 1 M = TM, T0 km = T k M, Tl 0M = T lm. If (x i ) are any local coordinates on U M and p U, the coordinate vectors i form a basis for T p M whose dual basis is {dx i }. Any tensor F Tl k(t pm) can be expresses in terms of this basis as F = F j1 j l j1 jl dx i1 dx i k. Definition. A tensor field on M is a smooth section of smoe tensor bundle T k l M. Covariant 1-tensor fields are covector fields. A 0-tensor field is a continuous real-valued function. Lemma 2. Let M be a smooth manifold, and let σ : M T k M be a rough section. The following are equivalent: (1) σ is smooth. (2) In any smooth coordinate chart, the component functions of σ are smooth. (3) If X 1,,X k are smooth vector fields defined on any subset U M, then the function σ(x 1,,X k ):U R, defined by σ(x 1,,X k )(p) =σ p (X 1 p,,x k p ), is smooth. Lemma 3. Let M be a smooth manifold, and suppose σ T k (M), τ T l (M), and f C (M). Then fσ and σ τ are also smooth tensor fields, whose components in any smooth local coordinate chart are (fσ) i1,,i k =fσ i1,,i k, (σ τ) i1,,i k+l =σ i1,,i k τ ik+1,,i k+l.
Pullbacks Just like smooth covector fields, smooth covariant tensor fields can be pulled back by smooth maps to yield smooth tensor fields. Definition. If F : M N is a smooth map, for each integer k 0 and each p M, we obtain a map F : T k (T F (p) N) T k (T p M) called the pullback by (F S)(X 1,,X k )=S(F X 1, F X k ). 5 Proposition 4 (Properties of Tensor Pullbacks). Suppose F : M N and G : N P are smooth maps, p M, S T k (T F (p) N), and T T l (T F (p) N). (a) F : T k (T F (p) N) T l (T p M) is linear over R. (b) F (S T )=F S F T. (c) (G F ) = F G : T k (T G F (p) P ) T k (T p M). (d) (Id N ) S = S. (e) F : T k N T k M is a smooth bundle map. Observe that properties (c), (d), and (e) imply that the assigments M T k M and R F yield a contravariant functor from the category of smooth manifolds and smooth maps to category of smooth bundles and smooth bundle maps. Definition. If F : M N is smooth, for any smooth covariant k-tensor field σ on N, we define a k-tensor field F σ on M, called the pullback of σ by F,by (F σ)=f (σ F (p) ). Write this in terms of its action on tangent vectors: If X 1,,X k T p M, then (F σ) p (X 1,,X k )=σ F (p) (F X 1, F X k ). If f is a smooth real-valued function (i.e. a smooth 0-tensor field) and σ is a smooth k-tensor field, then it is consistent with our definition to interpret f σ as fσ, and F f as f F. With these interpretations, propaerty (a) of this proposition is really a special case of (b). Proposition 5 (Properties of Tensor Field Pullbacks). Let F : M N and G : N P be smooth maps, σ T k (N), τ T l (N), and f C (N). (a) F (fσ)=(f F )F σ. (b) F (σ τ) =F σ F τ. (c) F σ is a smooth tensor field. (d) F : C k (N) T k (M) is linear over R. (e) (G F ) = F G. (f) (Id N ) σ = σ. Corollary 6. Let F : M N be a smooth map, and let σ T k (N). F (σ j1 j k dy j1 y j k )=(σ j1 j k F )d(y j1 F ) d(y j k F ). In words, whenever you see y j in the expression for σ, just substitute the jth component function of F and expand.
6 In general, there is neither a pushforward nor a pullback operation for mixed tensor field. However, in the special case of a diffeomorphism, tensor fields of any variance can be pushed forward and pulled back at will. Proposition. Suppose F : M N is a diffeomorphism. For any pair of nonnegative integers k, l, there are smooth bundle isomorphisms F : Tl km T l kn and F : Tl kn T l km F S(X 1,,X k,ω 1,,ω l )=S(F 1 X 1,,F 1 X k,f ω 1,,F ω l ), F S(X 1,,X k,ω 1,,ω l )=S(F X 1,,F X k,f 1 ω 1,,F 1 ω l ).