The Tensor Bundles

Subjects: Differential Geometry
Links: Vector Bundles on Smooth Manifolds, Tensor Product of Linear Functions, The Tangent Bundle, The Cotangent Bundle Local and Global Sections of Vector Bundles, Covector Fields on Smooth Manifolds

Def: Let $M$ be a smooth manifold with or without boundary. We define the bundle of covariant $k$-tensors on $M$ by $$T^k T^M := \coprod_{p\in M} T^k(T^p M) = \coprod{p\in M} (T_p^*M)^{\otimes k}. $$Analogously, we define the bundle of contravariant $k$-tensors by $$T^k TM := \coprod_{p\in M} T^k(T_p M) = \coprod_{p\in M} (T_pM)^{\otimes k},$$and the bundle of mixed tensors of type $(k,l)$ by $$T^{(k, l)} TM := \coprod_{p\in M} T^{(k, l)} (T_p M) = \coprod_{p\in M} \mathcal T^k_l (T_p M). $$

Any one of these bundle is called a tensor bundle over $M$. A section of a tensor bundle is called a (covariant, contravariant, or mixed) tensor field on $M$. A smooth tensor field is a section that is smooth in the usual sense of smooth sections of vector bundles.

Prop: The spaces of smooth sections of these tensor bundles, $\mathrm{\Gamma }(T^k{T}^{\ast }M)$, $\mathrm{\Gamma }(T^{k}TM)$ and $\mathrm{\Gamma }(T^{(k,l)}TM)$, are infinite-dimensional vector spaces over $\mathbb{R}$, and modules over ${\mathcal{C}}^{\mathrm{\infty }}(M)$.

In any smooth local coordinates $(x^i)$, sections of these bundles can be written as $$A = \begin{cases} A_{i_1,\dots, i_k} dx^{i_1}\otimes \dots\otimes dx^{i_k}, & A\in \Gamma(T^k T^* M); \ A ^{i_1,\dots, i_k} \dfrac{\partial}{\partial x^{i_1}}\otimes \dots \otimes\dfrac{\partial}{\partial x^{i_k}}, & A\in \Gamma(T^kTM); \
A ^{i_1,\dots, i_k}_{j_1,\dots, j_l} \dfrac{\partial}{\partial x^{i_1}}\otimes \dots \otimes\dfrac{\partial}{\partial x^{i_k}} \otimes dx^{j_1}\otimes \dots\otimes dx^{j_l}, & A\in \Gamma(T^{(k, l)}TM).
\end{cases} $$The functions $A_{i_1,\dots i_k}$, $A^{i_1,\dots ,i_k}$, or $A_{j_1,\dots ,j_l}^{i_1,\dots ,i_k}$ are called the component functions of $A$ in the chosen coordinates.

Because smooth covariant tensor fields occupy most of our attention, we adopt the following shorthand notation for the space of all smooth covariant $k$-tensor fields: $$\mathcal T^k (M) := \Gamma(T^k T^* M). $$
Smoothness Criteria for Tenser Fields: Let $M$ be a smooth manifold with or without boundary, and let $A:M\to T^k{T}^{\ast }M$ be a rough section. The following are equivalent.

• $A$ is smooth.
• In every smooth coordinate chart, the component functions of $A$ are smooth.
• Each point of $M$ is contained in some coordinate chart in which $A$ has smooth component functions.
• If $X_1,\dots ,X_k\in \mathfrak{X}(M)$, then the function $A(X_1,\dots ,X_k):M\to \mathbb{R}$, defined by $$A(X_1,\dots, X_k)(p) = A_p(X_1|_p,\dots, X_k|_p), $$is smooth.
• Whenever $X_1,\dots ,X_k$ are smooth vector fields defined on some open subset $U\subseteq M$, the function $A(X_1,\dots ,X_k)$ is smooth on $U$.

Prop: Suppose $M$ is a smooth manifold with or without boundary, $A\in {\mathcal{T}}^k(M)$, $B\in {\mathcal{T}}^l(M)$ and $f\in {\mathcal{C}}^{\mathrm{\infty }}(M)$. Then $fA$ and $A\otimes B$ are also smooth tensor fields, whose components in any smooth local coordinate chart are $$\begin{align*} (fA){i_1,\dots, i_k} &= fA{i_1,\dots, i_k}, \ (A \otimes B){i_1, \dots, i{k+l}} &= A_{i_1,\dots, i_k} B_{i_{k+1},\dots, i_{k+l}}.\end{align*} $$
Tensor Characterisation Lemma: A map $${\cal A}: {\frak X}(M)^k \to \mathcal C^\infty(M), $$is induced by a smooth covariant $k$-tensor field iff it is a multilinear over ${\mathcal{C}}^{\mathrm{\infty }}(M)$.

Def: A symmetric tensor field on a manifold with or without boundary is simply a covariant tensor field whose value at each point is a symmetric tensor. The symmetric product of two or more tensor field is defined pointwise, just like the tensor product.

Pullbacks of Tensor Fields

Just like covector fields, covariant tensor fields can be pulled back by a smooth map to yield a tensor field on the domain.

Suppose $F:M\to N$ is a smooth map. For any point $p\in M$ and any $k$-tensor $\alpha \in T^k(T_{F(p)}^{\ast }N)$, we define a tensor $d{F}_p^{\ast }(\alpha )\in T^k(T_p^{\ast }M)$, called the pullback of $\alpha$ by $F$ at $p$, by $$dF^_p(\alpha)(v_1,\dots, v_k) = \alpha (dF_p(v_1),\dots, dF_p(v_k)) $$for any $v_1,\dots ,v_k\in T_{p}M$. If $A$ is a covariant $k$-tensor field on $N$, we define a rough $k$-tensor field $F^{\ast }A$ on $M$, called the pullback of $A$ by $F$, by $$(F^A)_p := dF^p(A().$$This tensor fields acts on vectors $v_1,\dots ,v_k\in T_{p}M$ by $$(F^ A)p (v_1,\dots, v_k) = A(dF_p(v_1),\dots, dF_p(v_k)). $$
Properties of Tensor Pullbacks: Suppose $F:M\to N$ and $G:N\to P$ are smooth maps, and $A$ and $B$ are covariant tensor fields on $N$, and $f$ is a real-valued function on $N$.

• $F^{\ast }(fB)=(f\circ F)F^{\ast }B$.
• $F^{\ast }(A\otimes B)=F^{\ast }A\otimes F^{\ast }B$.
• $F^{\ast }(A+B)=F^{\ast }A+F^{\ast }B$.
• $F^{\ast }B$ is a continuous tensor field, and is smooth if $B$ is smooth.
• $(G\circ F{)}^{\ast }B=F^{\ast }(G^{\ast }B)$.
• $({\text{id}}_N{)}^{\ast }B=B$.
• If $A$ and $B$ are symmetric tensor fields on $N$, then $F^{\ast }A$ and $F^{\ast }B$ are symmetric, and $F^{\ast }(AB)=(F^{\ast }A)(F^{\ast }B)$.

Cor: Let $F:M\to N$ be smooth, and let $B$ be a covariant $k$-tensor field on $N$. If $p\in M$ and $(y^i)$ are smooth coordinates for $N$ on a neighbourhood of $F(p)$, then $F^{\ast }B$ has the following expression in a neighbourhood of $p$: $$F^(B_{i_1,\dots, i_k} dy^{i_1}\otimes \dots \otimes dy^{i_k}) = (B_{i_1,\dots, i_j} \circ F) d(y^{i_1}\circ F) \otimes d(y^{i_k}\circ F).$$
Cor: Let $M$ be a smooth $n$-manifold and let $A$ be a smooth tensor field on $M$. If $(U,(x^i))$ and $(V,(y^j))$ are overlapping smooth charts on $M$, we can write $$\begin{align}
A &= A ^{i_1,\dots, i_k}{j_1,\dots, j_l} \dfrac{\partial}{\partial x^{i_1}}\otimes \dots \otimes\dfrac{\partial}{\partial x^{i_k}} \otimes dx^{j_1}\otimes \dots\otimes dx^{j_l} \
&= \widetilde A ^{i_1,\dots, i_k}{j_1,\dots, j_l} \dfrac{\partial}{\partial y^{i_1}}\otimes \dots \otimes\dfrac{\partial}{\partial y^{i_k}} \otimes dy^{j_1}\otimes \dots\otimes dy^{j_l}\end{align*}.$$We can calculate the relation ship between $A_{j_1,\dots ,j_l}^{i_1,\dots ,i_k}$ and ${\tilde{A}}_{j_1,\dots ,j_l}^{i_1,\dots ,i_k}$, as $$\widetilde A ^{i_1,\dots, i_k}{j_1,\dots, j_l} = A^{m_1,\dots, m_k}{n_1,\dots, n_l} \frac{\partial y^{i_1}}{\partial x^{m_1}}\cdots \frac{\partial y^{i_k}}{\partial x^{m_k}} \frac{\partial x^{n_1}}{\partial y^{j_1}} \dots \frac{\partial x^{n_l}}{\partial y^{j_l}}.$$
Prop: For every pair of nonnegative integers $k,l$ and every diffeomorphism $F:M\to N$ are the pushforward and pullback isomorphisms.

• $F_{\ast }:\mathrm{\Gamma }(T^{(k,l)}TM)\to \mathrm{\Gamma }(T^{(k,l)}TN),$
• $F_{\ast }:\mathrm{\Gamma }(T^{(k,l)}TN)\to \mathrm{\Gamma }(T^{(k,l)}TM)$,
  such that $F^{\ast }$ agrees with the usual pullback on covariant tensor fields, $F_{\ast }$ agrees with the usual pushforward on contravariant $1$-tensor fields, and the following conditions are satisfied:
• $F_{\ast }=(F^{\ast }{)}^{-1}$.
• $F^{\ast }(A\otimes B)=F^{\ast }A\otimes F^{\ast }B$.
• $(F\circ G{)}_{\ast }=F_{\ast }\circ G_{\ast }$.
• $(F\circ G{)}^{\ast }=G^{\ast }\circ F^{\ast }$.
• $({\text{id}}_M{)}^{\ast }=({\text{id}}_M{)}_{\ast }=\text{id}:\mathrm{\Gamma }(T^{(k,l)}TM)\to \mathrm{\Gamma }(T^{(k,l)}TM)$.
• $F^{\ast }(A(X_1,\dots ,X_k))=F^{\ast }A(F_{\ast }^{-1}(X_1),\dots ,F_{\ast }^{-1}(X_k))$ for $A\in {\mathcal{T}}^k(N)$ and $X_1,\dots ,X_k\in \mathfrak{X}(N)$.