Tensors

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Definitions

There are two ways we can think of a rank (k,l) tensor on a vector space V. The first is as a multilinear map from

$LaTeX: \underbrace{V^* \times \ldots \times V^*}_{l~times} \times \underbrace{V \times \ldots \times V}_{k~times} \to R$ .

Alternatively, we can think of a rank (p,q) tensor as an element of the space

$LaTeX: \underbrace{V \otimes \ldots \otimes V}_{l~times} \otimes \underbrace{V^* \otimes \ldots \otimes V^*}_{k~times}.$

The relationship between these two definitions is that an element $LaTeX: v_1 \otimes \ldots \otimes v_l \otimes \phi^1 \otimes \ldots \otimes \phi^k$ acts multilinearly on $LaTeX: \underbrace{V^* \times \ldots \times V^*}_{l~times} \times \underbrace{V \times \ldots \times V}_{k~times}$ as

$LaTeX: v_1 \otimes \ldots \otimes v_l \otimes \phi^1 \otimes \ldots \otimes \phi^k (\psi^1, \ldots, \psi^q, w_1, \ldots, w_k) = \psi^1(v_1) \ldots \psi^l(v_l) \phi^1(w_1) \ldots \phi^k(w_k) \in R$ .

We denote the vector space of all tensors of rank (k,l) on the vector space V by $LaTeX: T_l^k(V)$ . Given a basis $LaTeX: \{e_1, \ldots, en \}$ for V with corresponding dual basis $LaTeX: \{f^1,\ldots, f^n\}$ , we can write any element G in $LaTeX: T_q^p(V)$ (uniquely) as

$LaTeX: G = G_{j_1 \ldots j_k}^{i_1 \ldots i_l} e_{i_1} \otimes \ldots \otimes e_{i_l} \otimes f^{j_1} \otimes f^{j_k}.$

Notice that

$LaTeX: G_{j_1 \ldots j_k}^{i_1 \ldots i_l} e_{i_1} = G(f^{i_1}, \ldots, f^{i_l}, e_{j_1} \ldots e_{j_k}).$

The identification of (1,1) tensors with End(V)

An important identification is the space $LaTeX: T_1^1(V)$ with End(V), the vector space of all linear transformations from V to itself. The isomorphism $LaTeX: \Gamma: End(V) \to T_1^1$ is given by $LaTeX: \Gamma(T)(\phi, v) = \phi(T(v))$ . In a basis $LaTeX: \{e_1, \ldots, e_n\}$ for V with corresponding dual basis $LaTeX: \{f^1, \ldots, f^n\}$ , the inverse map takes the form $LaTeX: (\Gamma^{-1}(G))(v) = G(f^i, v)e_i$ . Indeed, we have that

$LaTeX: \Gamma(\Gamma^{-1}(G))(\phi, v) = \phi((\Gamma^{-1}(G))(v)) = \phi(G(f^i,v)e_i) = G(f^i,v)\phi(e_i) = G(\phi(e_i)f^i,v) = G(\phi, v)$

and

$LaTeX: \Gamma^{-1}(\Gamma(T))(v) = (\Gamma(T)(f^i,v)) e_i = f^i(T(v)) e_i = T(v).$ .

Note that $LaTeX: tr(\Gamma^{-1}(G)) = f^i(\Gamma^{-1}(G)(e_i)) = f^i(G(f^j,e_i) e_j) = G(f^j,e_i) f^i(e_j) = G(f^i, e_i)$ and, in particular, $LaTeX: tr(\Gamma^{-1}(v \otimes \phi)) = v \otimes \phi (f^i, e_i) = v(f_i) \phi(e_i) = v^i \phi_i = \phi(v)$ .

Tensor Contraction

We can use this isomorphism to define a natural map, called contraction, from $LaTeX: T^{k+1}_{l+1} \to T_q^p$ . Given a (k+1,l+1) tensor F we can fix p covectors and q vectors to get a (1,1) tensor. By the above identification, we can view this as an element of End(V) and thus we can define the action of the contraction of F on $LaTeX: (\phi^1, \ldots, \phi^l, v_1, \ldots, v_k)$ to be

$LaTeX: tr( \Gamma^{-1}(F(\phi^1, \ldots, \phi^l, \cdot, v_1, \ldots, v_k, \cdot))).$

In a basis $LaTeX: \{e_i, \ldots, e_n\}$ for V, we have that

$LaTeX: tr( \Gamma^{-1}(F(\phi^1, \ldots, \phi^l, \cdot, v_1, \ldots, v_k, \cdot)) = F(\phi^1, \ldots, \phi^l, f^i, v_1, \ldots, v_k, e_i).$

Therefore, the components of the contraction of F are $LaTeX: F_{j_1 \ldots j_k}^{i_1 \ldots i_l} = F(f^{i_1}, \ldots, f^{i_l}, f^m, e_{j_1}, \ldots, e_{j_k}, e_m) = F_{j_1 \ldots j_p m}^{i_1 \ldots i_l m}.$

Notice that, by a computation in the previous section, the contraction of the tensor

$LaTeX: v_1 \otimes \ldots \otimes v_{l+1} \otimes \phi^1 \otimes \ldots \otimes \phi^{k+1}$

is the tensor

$LaTeX: \phi^{k+1}(v_{l+1}) v_1 \otimes \ldots \otimes v_l \otimes \phi^1 \otimes \ldots \otimes \phi^k.$

Abstract Index Notation

Tensor Fields

A rank (k,l) tensor field is a function that smoothly assigns to each point $LaTeX: p\in M$ an element of $LaTeX: T_l^k(T_p M)$ . More formally, we can form a vector bundle called the tensor bundle whose fiber over p is $LaTeX: \{p\} \times T_l^k(T_p M)$ . This vector bundle is given a smooth structure (and corresponding topology) in the following way: if U is a coordinate chart for p with coordinates $LaTeX: \phi = (x^i)$ , then any element of $LaTeX: \pi^{-1}(U)$ can be written as

$LaTeX: F^{i_1 \ldots i_q}_{j_1 \ldots j_p} \partial_{i_1} \vert_p \otimes \ldots \otimes \partial_{i_q} \vert_p \otimes dx^{j_1} \vert_p \otimes \ldots \otimes dx^{j_p} \vert_p.$

Thus we take the coordinate domains to be of the form $LaTeX: \pi^{-1}(U)$ with coordinate maps

$LaTeX: F^{i_1 \ldots i_l}_{j_1 \ldots j_k} \partial_{i_1} \vert_p \otimes \ldots \otimes \partial_{i_l} \vert_p \otimes dx^{j_1} \vert_p \otimes \ldots \otimes dx^{j_k} \vert_p \mapsto (\phi(p), F^{i_1 \ldots i_l}_{j_1 \ldots j_k})_{i_r,j_s=1,\ldots,n} \in \phi(U) \times R^{n^{k + l}}.$

We will denote the rank (k,l) tensor bundle by $LaTeX: T_l^k(M)$ and the set of all smooth sections of this bundle by $LaTeX: \mathcal{T}_l^k(M)$ .

An equivalent way to view a smooth tensor field F of rank (k,l) is as a $LaTeX: C^\infty(M)$ linear map $LaTeX: \mathcal{T}(M) \times \ldots \times \mathcal{T}(M) \times \mathcal{T}^1(M) \times \ldots \times \mathcal{T}^1(M) \to C^\infty(M)$ where