Differentiable Manifolds/What is a manifold?

In this section, the important concepts of manifolds shall be introduced.

Charts, compatibility of charts, atlases and manifolds

In this subsection, we define a manifold and all the things which are necessary to define it. It's a bit lengthy for a definition, but manifolds are such an important concept in mathematics that it's far more than worth it.

Definition 1.1:

Let $M$ be a topological space, let $d\in \mathbb {N}$ be a natural number, and let $O\subseteq M$ be open. We call a function $\phi :O\to \mathbb {R} ^{d}$ a chart iff $\phi :O\to \phi (O)$ is a homeomorphism and $\phi (O)$ is open in $\mathbb {R} ^{d}$.

Definition 1.2:

Let $M$ be a topological space, let $d\in \mathbb {N}$ be a natural number, let $O,U\subseteq M$ be open, let $\phi :O\to \mathbb {R} ^{d}$ and $\theta :U\to \mathbb {R} ^{d}$ be two charts, and let $n\in \mathbb {N} _{0}\cup \{\infty \}$. We call the two charts compatible of class ${\mathcal {C}}^{n}$ iff either

$U\cap O=\emptyset$

or

both maps

$\theta |_{U\cap O}\circ {\phi |_{U\cap O}}^{-1}:\phi (U\cap O)\to \theta (U\cap O)$

and

$\phi |_{U\cap O}\circ {\theta |_{U\cap O}}^{-1}:\theta (U\cap O)\to \phi (U\cap O)$

are contained in ${\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} ^{d})$.

Definition 1.3:

Let $M$ be a topological space, let $d\in \mathbb {N}$, let $n\in \mathbb {N} _{0}\cup \{\infty \}$, let $\{O_{\upsilon }|\upsilon \in \Upsilon \}$, where $\Upsilon$ is a set, be a set of open subsets of $M$, and let $\{\phi _{\upsilon }:O_{\upsilon }\to \mathbb {R} ^{d}|\upsilon \in \Upsilon \}$ be an according set of charts. We call the set $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ an atlas of class ${\mathcal {C}}^{n}$ of $M$ iff both

• for all $p\in M$, there exists an $(O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ such that $p\in O$ and
• for all $(O,\phi ),(U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ the charts $\phi$ and $\theta$ are compatible of class ${\mathcal {C}}^{n}$.

Definition 1.4:

Let $M$ be a topological space, let $d\in \mathbb {N}$ and let $n\in \mathbb {N} _{0}\cup \{\infty \}$. Together with an atlas $\{O_{\upsilon }|\upsilon \in \Upsilon \}$ of $M$ of class ${\mathcal {C}}^{n}$, we call $M$ a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$.

Differentiable functions on manifolds

In this subsection, we shall define what differentiable maps, which map from a manifold or to a manifold or both, are.

Definition 1.5:

Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, let $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ be an atlas of it, and let $k\in \mathbb {N} \cup \{\infty \}$, and let $U\subseteq M$ be open. We call a function $\varphi :U\to \mathbb {R}$ differentiable of class ${\mathcal {C}}^{k}$ iff for all $(O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ the function

$\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}:\phi (O)\to \mathbb {R}$

is contained in ${\mathcal {C}}^{k}(\mathbb {R} ^{d},\mathbb {R} )$.

We write ${\mathcal {C}}^{k}(M)$ for the set of all real-valued differentiable functions of class ${\mathcal {C}}^{k}$ from any open subset of $M$ to $\mathbb {R}$. Further, we write ${\mathcal {C}}^{0}(M)$ for the set of continuous functions from any open subset of $M$ to $\mathbb {R}$ (remember that both $M$ and $\mathbb {R}$ are topological spaces, which is why continuity is defined for functions from one of them to the other).

On this set ${\mathcal {C}}^{k}(M)$, we define addition and multiplication as follows: Let $U,O\subseteq M$ be open and $\varphi :O\to \mathbb {R}$, $\vartheta :U\to \mathbb {R}$ be two differentiable functions of class ${\mathcal {C}}^{k}$. We define

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

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

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

and, if $\vartheta$ is never zero,

${\frac {\varphi }{\vartheta }}:O\cap U\to \mathbb {R} ,\left({\frac {\varphi }{\vartheta }}\right)(p)={\frac {\varphi (p)}{\vartheta (p)}}$

Instead of writing $\varphi \cdot \vartheta$, we will in the following write $\varphi \vartheta$; just omitting the dot. This is often also done for the multiplication of variables (for instance $xy$ stands for $x\cdot y$ if $x,y\in \mathbb {R}$).

Definition 1.6:

Let $M$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, let $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ be an atlas of it, let $k\in \mathbb {N} \cup \{\infty \}$ and let $I\subseteq \mathbb {R}$. We call a function $\gamma :I\to M$ a differentiable curve of class ${\mathcal {C}}^{k}$ iff for all $(O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ the function

$\phi \circ \gamma |_{\gamma ^{-1}(O)}:\gamma ^{-1}(O)\to \mathbb {R} ^{d}$

is contained in ${\mathcal {C}}^{k}(\mathbb {R} ,\mathbb {R} ^{d})$.

Definition 1.7:

Let $M,N$ be manifolds of dimensions $d,b\in \mathbb {N}$ respectively, let $k\in \mathbb {N} \cup \{\infty \}$ and let $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$, $\{(U_{\kappa },\theta _{\kappa })|\kappa \in \mathrm {K} \}$ be atlases of $M$ and $N$ respectively. We call a function $\psi :M\to N$ differentiable of class ${\mathcal {C}}^{k}$ iff for all $(O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ and all $(U,\theta )\in \{(U_{\kappa },\theta _{\kappa })|\kappa \in \mathrm {K} \}$ either

$O\cap \psi ^{-1}(U)=\emptyset$

or the function

$\theta \circ \psi \circ \phi ^{-1}|_{\phi (O\cap \psi ^{-1}(U))}:\phi (O\cap \psi ^{-1}(U))\to \mathbb {R} ^{b}$

is contained in ${\mathcal {C}}^{k}(\mathbb {R} ^{d},\mathbb {R} ^{b})$.

Tangent vectors, tangent spaces and tangent bundles

Tangents, in the classical sense, are lines which touch a geometrical object at exactly one point. The following definition of a tangent of a manifold, in this context called tangent vector to a manifold, is somewhat strange.

Definition 1.8:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, and let $p\in M$. A tangent vector at $p$ is a function $\mathbf {V} _{p}:{\mathcal {C}}^{n}(M)\to \mathbb {R}$ such that for all $c\in \mathbb {R}$ and $\varphi ,\vartheta \in {\mathcal {C}}^{n}(M)$ the following three rules hold:

1. $\mathbf {V} _{p}(\varphi +c\vartheta )=\mathbf {V} _{p}(\varphi )+c\mathbf {V} _{p}(\vartheta )$ whenever $\varphi$ and $\vartheta$ are both defined at $p$ (linearity)
2. $\mathbf {V} _{p}(\varphi \vartheta )=\varphi (p)\mathbf {V} _{p}(\vartheta )+\vartheta (p)\mathbf {V} _{p}(\varphi )$ whenever $\varphi$ and $\vartheta$ are both defined at $p$ (product rule)
3. if $\varphi$ is not defined at $p$ (i. e. $\varphi$ is a function from $O$ to $\mathbb {R}$ such that $p\notin O$), then $\mathbf {V} _{p}(\varphi )=0$.

Definition 1.9:

Let $M$ be a manifold, and let $p\in M$. The tangent space of $M$ in $p$, which we shall denote by $T_{p}M$, is defined to be the vector space of all tangent vectors at $p$ with the scalar-vector multiplication

$(c\mathbf {V} _{p})(\varphi ):=c\mathbf {V} _{p}(\varphi )$,

the vector-vector addition

$(\mathbf {V} _{p}+\mathbf {W} _{p})(\varphi ):=\mathbf {V} _{p}(\varphi )+\mathbf {W} _{p}(\varphi )$

and the zero element

$\mathbf {0} _{p}(\varphi ):=0$.

Definition 1.10:

Let $M$ be a manifold. The tangent bundle of $M$, denoted by $TM$, is defined as follows:

$TM:=\bigcup _{p\in M}T_{p}M$

Cotangent vectors, cotangent spaces and cotangent bundles

Definition 1.11:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, and let $p\in M$. A linear function from $T_{p}M$ to $\mathbb {R}$ is called cotangent vector at $p$. One standard symbol for a cotangent vector at $p$ is $\alpha _{p}$.

Definition 1.12:

Let $M$ be a manifold, and let $p\in M$. The cotangent space of $M$ in $p$, which we shall denote by $T_{p}M^{*}$, is defined to be the vector space of all cotangent vectors at $p$ with the scalar-vector multiplication

$(c\alpha _{p})(v_{p}):=c\alpha _{p}(v_{p})$,

the vector-vector addition

$(\alpha _{p}+\beta _{p})(v_{p}):=\alpha _{p}(v_{p})+\beta _{p}(v_{p})$

and the zero element

$0_{p}(v_{p}):=0$.

Definition 1.13:

Let $M$ be a manifold. The cotangent bundle of $M$, denoted by $TM^{*}$, is defined as follows:

$TM^{*}:=\bigcup _{p\in M}T_{p}M^{*}$

Tensors and the tensor product

Definition 1.14:

Let $V$ be a vector space, $V^{*}$ its dual space and let $k,m\in \mathbb {N} _{0}$. We call a multilinear function

$T:\overbrace {V^{*}\times \cdots \times V^{*}} ^{k{\text{ times}}}\times \overbrace {V\times \cdots \times V} ^{m{\text{ times}}}\to \mathbb {R}$

a $(k,m)$ tensor.

Definition 1.15: Let $V$ be a vector space, let $V^{*}$ be its dual space, let $k,m,j,l\in \mathbb {N} _{0}$, let $T$ be a $(k,m)$ tensor and let $S$ be a $(j,l)$ tensor. The tensor product of $T$ and $S$, denoted by $T\otimes S$, is defined to be the $(k+j,m+l)$ tensor given by

${\begin{aligned}T\otimes S&:~\overbrace {V^{*}\times \cdots \times V^{*}} ^{k+j{\text{ times}}}\times \overbrace {V\times \cdots \times V} ^{m+l{\text{ times}}}\to \mathbb {R} ,\\(T\otimes S)(\mathbf {v} _{1}^{*},\ldots ,\mathbf {v} _{k+j}^{*},\mathbf {v} _{1},\ldots ,\mathbf {v} _{m+l})&=T(\mathbf {v} _{1}^{*},\ldots ,\mathbf {v} _{k}^{*},\mathbf {v} _{1},\ldots ,\mathbf {v} _{m})~\cdot \\&~~~~S(\mathbf {v} _{k+1}^{*},\ldots ,\mathbf {v} _{k+j}^{*},\mathbf {v} _{m+1},\ldots ,\mathbf {v} _{m+l})\end{aligned}}$

We denote the set of all tensors with respect to $V$ by $T(V)$.

Sources

• Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.
• Lang, Serge (2002). Introduction to Differentiable Manifolds. New York: Springer. ISBN 0-387-95477-5.
• Rudolph, Gerd; Schmidt, Matthias (2013). Differential Geometry and Mathematical Physics. Netherlands: Springer. ISBN 978-94-007-5345-7.