# Differentiable Manifolds/What is a manifold?

 Differentiable Manifolds What is a manifold? Bases of tangent and cotangent spaces and the differentials →

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

Definition 1.2:

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

${\displaystyle U\cap O=\emptyset }$

or

both maps

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

and

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

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

Definition 1.3:

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

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

Definition 1.4:

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

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

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

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

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

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

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

and, if ${\displaystyle \vartheta }$ is never zero,

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

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

Definition 1.6:

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

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

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

Definition 1.7:

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

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

or the function

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

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

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

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

Definition 1.9:

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

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

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

and the zero element

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

Definition 1.10:

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

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

## Cotangent vectors, cotangent spaces and cotangent bundles

Definition 1.11:

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

Definition 1.12:

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

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

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

and the zero element

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

Definition 1.13:

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

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

## Tensors and the tensor product

Definition 1.14:

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

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

a ${\displaystyle (k,m)}$ tensor.

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

{\displaystyle {\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 ${\displaystyle V}$ by ${\displaystyle T(V)}$.

## 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.
 Differentiable Manifolds What is a manifold? Bases of tangent and cotangent spaces and the differentials →