Differentiable Manifolds/Vector fields, covector fields, the tensor algebra and tensor fields

In this section, the concepts of vector fields, covector fields and tensor fields shall be presented. We will also define what it means that one of those (vector field, covector field, tensor field) is differentiable. Then we will show how suitable restrictions of all these things can be written as sums of the bases of the respective spaces induced by a chart, and we will show a simultaneously sufficient and necessary condition of differentiability based on this sum expression.

Vector fields

Definitions 5.1:

Let $M$ be a manifold, let $O\subseteq M$ be open, and let

$\mathbf {X} :O\to TM$

be a function from $O$ to the tangent bundle of $M$, such that:

$\forall p\in O:\mathbf {X} (p)\in T_{p}M$

Then we call $\mathbf {X}$ a vector field on $O$.

If $\mathbf {X} :O\to TM$ and $\mathbf {Y} :U\to TM$ are two vector fields and $c\in \mathbb {R}$, we define

$\mathbf {X} +c\mathbf {Y} :O\cap U\to TM,(\mathbf {X} +c\mathbf {Y} )(p):=\mathbf {X} (p)+c\mathbf {Y} (p)$

The set of all vector fields on $M$ we denote by ${\mathfrak {X}}(M)$.

Definition 5.2:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, let $O\subseteq M$ be open, let $\mathbf {X}$ be a vector field on $O$ and let $0\leq k\leq n$. We call $\mathbf {X}$ differentiable of class ${\mathcal {C}}^{k}$ iff for all $\varphi :U\to \mathbb {R}$ in ${\mathcal {C}}^{k}(M)$ the function

$\mathbf {X} \varphi :O\cap U\to \mathbb {R} ,\mathbf {X} \varphi (p):=\mathbf {X} (p)(\varphi )$

is contained in ${\mathcal {C}}^{k}(M)$.

The set of all vector fields of class ${\mathcal {C}}^{k}$ is denoted by ${\mathfrak {X}}^{k}(M)$.

Lemma 5.4:

Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$ and $(O,\phi )$ be contained in its atlas. Then the vector fields

$p\mapsto \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p},j\in \{1,\ldots ,d\}$

are differentiable of class ${\mathcal {C}}^{n}$.

Proof:

Let $\varphi \in {\mathcal {C}}^{\infty }(M)$. Then we have:

$\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\varphi (p)=\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))$

Let now $(V,\theta )$ be another chart in the atlas of $M$. Then the function

$\left({\frac {\partial }{\partial \phi _{j}}}\right)\varphi {\big |}_{O\cap U\cap V}\circ \theta |_{O\cap U\cap V}^{-1}=\partial _{x_{j}}(\varphi \circ \phi ^{-1})\circ (\phi |_{O\cap U\cap V}\circ \theta |_{O\cap U\cap V}^{-1})$

is smooth, as the composition of smooth functions.$\Box$

If $M$ is a manifold of class ${\mathcal {C}}^{\infty }$, we even have that since $\mathbf {X} (p)$ is contained in $T_{p}M$ for all $p\in O$, if a chart around $q\in O$ is given by $\theta :U\to \theta (U)\subseteq \mathbb {R} ^{d}$, then for all $p\in O\cap U$

$\mathbf {X} (p)=\sum _{j=1}^{d}x_{j}(p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}~~~~~(*)$

, where

$x_{j}:U\cap O\to \mathbb {R}$

are functions from $U\cap O$ to $\mathbb {R}$. This follows from chapter 2, where we remarked based on two theorems of the section, that

$\left\{\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}$

is a basis of $T_{p}M$ for $p\in O$.

Theorem 5.5:

Let $\mathbf {X}$ be a vector field on the open subset $U$ of the $d$-dimensional ${\mathcal {C}}^{\infty }$ manifold $M$, and let $(O,\phi )$ be contained in the atlas of $M$. Then the vector field $\mathbf {X} |_{O\cap U}$ is differentiable of class ${\mathcal {C}}^{\infty }$ iff all the $x_{j},j\in \{1,\ldots ,d\}$, as defined by eq. $(*)$ above, are contained in ${\mathcal {C}}^{\infty }(M)$.

Proof:

1.) We prove that if all the $\mathbf {V} _{\phi ,j}$ defined by $(*)$ are contained in ${\mathcal {C}}^{\infty }(M)$, that then $\mathbf {V}$ is differentiable of class ${\mathcal {C}}^{\infty }$.

This is because if $\varphi$ is contained in ${\mathcal {C}}^{\infty }(M)$, then due to lemma 5.4 and theorem 2.24 all the summands of the function

$\sum _{j=1}^{d}\mathbf {V} _{\phi ,j}\left({\frac {\partial }{\partial \phi _{j}}}\right)\varphi$

are differentiable of class ${\mathcal {C}}^{\infty }$. Due to theorem 2.23 and induction, we have that the function itself is differentiable of class ${\mathcal {C}}^{\infty }$. Due to $(*)$, the function is identical to $\mathbf {V}$.

2.) We prove that if $\mathbf {V}$ is differentiable of class ${\mathcal {C}}^{\infty }$, then so are the $V_{\phi ,j}$ defined by $(*)$.

Due to lemma 2.3, if we write $\phi =(\phi _{1},\ldots ,\phi _{d})$, the functions $\phi _{k}:M\to \mathbb {R}$, $k\in \{1,\ldots ,d\}$ are contained in ${\mathcal {C}}^{\infty }(M)$.

By definition of the differentiability of class ${\mathcal {C}}^{\infty }$ of $\mathbf {V}$, we have that the functions

$\mathbf {V} \phi _{k},k\in \{1,\ldots ,d\}$

are contained in ${\mathcal {C}}^{\infty }(M)$. But due to $(*)$ and lemma 2.4, we have for all $p\in O\cap U$:

${\begin{aligned}\mathbf {V} \phi _{k}(p)&=\mathbf {V} _{\phi ,j}(p)\sum _{j=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})\\&=\mathbf {V} _{\phi ,j}(p)\end{aligned}}$

Hence:

$\mathbf {V} \phi _{k}=\mathbf {V} _{\phi ,j}$

, where since the two functions are equal and one of them is differentiable of class ${\mathcal {C}}^{\infty }$, both of them are differentiable of class ${\mathcal {C}}^{\infty }$.$\Box$

Covector fields

Definition 5.6:

Let $M$ be a manifold, let $O\subseteq M$ be open, and let

$\alpha :O\to TM^{*}$

be a function from $O$ to the cotangent bundle of $M$, such that:

$\forall p\in O:\alpha (p)\in T_{p}M^{*}$

Then we call $\alpha$ a covector field on $O$.

Definition 5.7:

Let $M$ be a manifold, let $O\subseteq M$ be open and let $\alpha :O\to TM^{*}$ be a covector field. Then we call $\alpha$ differentiable of class ${\mathcal {C}}^{k}$ iff for all vector fields $\mathbf {X} :U\to TM$ which are differentiable of class ${\mathcal {C}}^{k}$, we have that the function

$\alpha \mathbf {X} :O\cap U\to \mathbb {R} ,\alpha \mathbf {X} (p):=\alpha (p)(\mathbf {V} (p))$

is contained in ${\mathcal {C}}^{k}(M)$.

Lemma 5.9:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$ and $(O,\phi )$ be contained in its atlas. Then the covector fields

$p\mapsto (d\phi _{j}),j\in \{1,\ldots ,d\}$

are differentiable of class ${\mathcal {C}}^{n}$.

Proof:

Let $\mathbf {V}$ be differentiable of class ${\mathcal {C}}^{\infty }$, and let $j\in \{1,\ldots ,d\}$. Due to lemma 2.3, the function $\phi _{j}$ is differentiable of class ${\mathcal {C}}^{\infty }$. Since $\mathbf {V}$ is differentiable of class ${\mathcal {C}}^{\infty }$, it follows that

$\mathbf {V} \phi _{j}=d\phi _{j}\mathbf {V}$

is differentiable of class ${\mathcal {C}}^{\infty }$ (the latest equation follows from the definition of $d\phi _{j}$).$\Box$

If $M$ is a manifold of class ${\mathcal {C}}^{\infty }$, we even have that since $\alpha (p)$ is contained in $T_{p}M^{*}$ for all $p\in O$, if a chart around $q\in O$ is given by $\theta :U\to \theta (U)\subseteq \mathbb {R} ^{d}$, then for all $p\in O\cap U$

$\alpha (p)=\sum _{j=1}^{d}x_{j}^{*}(p)(d\varphi _{j})_{p}~~~~~(**)$

, where

$x_{j}^{*}:U\cap O\to \mathbb {R}$

are functions from $U\cap O$ to $\mathbb {R}$. This follows from chapter 2, where we remarked based on two theorems of the chapter, that

$\left\{(d\phi _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}$

is a basis of $T_{p}M^{*}$ for $p\in O$.

Theorem 5.10:

Let $\alpha :O\to TM^{*}$ be a covector field on the $d$-dimensional ${\mathcal {C}}^{\infty }$ manifold $M$. Then $\alpha$ is differentiable of class ${\mathcal {C}}^{\infty }$ iff all the $x_{j}^{*}$, $j\in \{1,\ldots ,d\}$, as defined in equation $(**)$, are contained in ${\mathcal {C}}^{\infty }(M)$.

Proof:

1.) We show that the differentiability of class ${\mathcal {C}}^{\infty }$ of the $\alpha _{\phi ,j}$ defined by $(**)$ implies the differentiability of $\alpha$.

Let $\mathbf {V}$ be a vector field on $M$ which is differentiable of class ${\mathcal {C}}^{\infty }$. Due to lemma 5.9 and theorem 2.24, all the summands of the function

$\sum _{j=1}^{d}\alpha _{\phi ,j}d\phi _{j}\mathbf {V}$

are contained in ${\mathcal {C}}^{\infty }(M)$. Therefore, due to theorem 2.23 and induction, also the function itself is contained in ${\mathcal {C}}^{\infty }(M)$. But due to $(**)$, the function is equal to $\alpha \mathbf {V}$.

2.) We show that if $\alpha$ is differentiable, then so are the $\alpha _{\phi ,j}$, $j\in \{1,\ldots ,d\}$ defined by $(**)$.

Due to lemma 5.4, we have that for $k\in \{1,\ldots ,d\}$, the vector field $\left({\frac {\partial }{\partial \phi _{k}}}\right)$ is differentiable of ${\mathcal {C}}^{\infty }$. Hence, due to the differentiability of $\alpha$, the function

$\alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)$

is contained in ${\mathcal {C}}^{\infty }(M)$. But due to $(**)$, we have

${\begin{aligned}\alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)(p)&=\sum _{j=1}^{d}\alpha _{\phi ,j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)\\&=\alpha _{\phi ,k}(p)\end{aligned}}$

Hence:

$\alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)=\alpha _{\phi ,k}$

and hence $\alpha _{\phi ,k}\in {\mathcal {C}}^{\infty }(M)$.$\Box$

The tensor algebra

Definition 5.9:

By an algebra, one often means a vector space $A$ such that there is a function $\cdot :A\times A\to A$ which is bilinear, i. e. satisfies for all $\mathbf {u} ,\mathbf {v} ,\mathbf {w} \in A$ and $c\in \mathbb {R}$:

$\mathbf {u} \cdot (\mathbf {v} +c\mathbf {w} )=\mathbf {u} \cdot \mathbf {v} +c\mathbf {u} \cdot \mathbf {w}$ and $(\mathbf {u} +c\mathbf {v} )\cdot \mathbf {w} =\mathbf {u} \cdot \mathbf {w} +c\mathbf {v} \cdot \mathbf {w}$

Definition 5.11:

Let $V$ be a vector space, and $V^{*}$ its dual space.

Tensor fields

Definition 5.?: