Tangent space

Definition via tangent curves

In the embedded-manifold picture, a tangent vector at a point ${\displaystyle x}$ is thought of as the velocity of a curve passing through the point ${\displaystyle x}$. We can therefore define a tangent vector as an equivalence class of curves passing through ${\displaystyle x}$ while being tangent to each other at ${\displaystyle x}$.

Suppose that ${\displaystyle M}$ is a ${\displaystyle C^{k}}$ differentiable manifold (with smoothness ${\displaystyle k\geq 1}$) and that ${\displaystyle x\in M}$. Pick a coordinate chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$, where ${\displaystyle U}$ is an open subset of ${\displaystyle M}$ containing ${\displaystyle x}$. Suppose further that two curves ${\displaystyle \gamma _{1},\gamma _{2}:(-1,1)\to M}$ with ${\displaystyle {\gamma _{1}}(0)=x={\gamma _{2}}(0)}$ are given such that both ${\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}}$ are differentiable in the ordinary sense (we call these differentiable curves initialized at ${\displaystyle x}$). Then ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$ are said to be equivalent at ${\displaystyle 0}$ if and only if the derivatives of ${\displaystyle \varphi \circ \gamma _{1}}$ and ${\displaystyle \varphi \circ \gamma _{2}}$ at ${\displaystyle 0}$ coincide. This defines an equivalence relation on the set of all differentiable curves initialized at ${\displaystyle x}$, and equivalence classes of such curves are known as tangent vectors of ${\displaystyle M}$ at ${\displaystyle x}$. The equivalence class of any such curve ${\displaystyle \gamma }$ is denoted by ${\displaystyle \gamma '(0)}$. The tangent space of ${\displaystyle M}$ at ${\displaystyle x}$, denoted by ${\displaystyle T_{x}M}$, is then defined as the set of all tangent vectors at ${\displaystyle x}$; it does not depend on the choice of coordinate chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$.

To define vector-space operations on ${\displaystyle T_{x}M}$, we use a chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$ and define a map ${\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} ^{n}}$ by ${\textstyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[(\varphi \circ \gamma )(t)]\right|_{t=0},}$ where ${\displaystyle \gamma \in \gamma '(0)}$. The map ${\displaystyle \mathrm {d} {\varphi }_{x}}$ turns out to be bijective and may be used to transfer the vector-space operations on ${\displaystyle \mathbb {R} ^{n}}$ over to ${\displaystyle T_{x}M}$, thus turning the latter set into an ${\displaystyle n}$-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$ and the curve ${\displaystyle \gamma }$ being used, and in fact it does not.

Definition via derivations

Suppose now that ${\displaystyle M}$ is a ${\displaystyle C^{\infty }}$ manifold. A real-valued function ${\displaystyle f:M\to \mathbb {R} }$ is said to belong to ${\displaystyle {C^{\infty }}(M)}$ if and only if for every coordinate chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$, the map ${\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbb {R} ^{n}\to \mathbb {R} }$ is infinitely differentiable. Note that ${\displaystyle {C^{\infty }}(M)}$ is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.

A derivation at ${\displaystyle x\in M}$ is defined as a linear map ${\displaystyle D:{C^{\infty }}(M)\to \mathbb {R} }$ that satisfies the Leibniz identity $${\displaystyle \forall f,g\in {C^{\infty }}(M):\qquad D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g),}$$ which is modeled on the product rule of calculus.

(For every identically constant function ${\displaystyle f={\text{const}},}$ it follows that ${\displaystyle D(f)=0}$).

Denote ${\displaystyle T_{x}M}$ the set of all derivations at ${\displaystyle x.}$ Setting

• ${\displaystyle (D_{1}+D_{2})(f):={D}_{1}(f)+{D}_{2}(f)}$ and
• ${\displaystyle (\lambda \cdot D)(f):=\lambda \cdot D(f)}$

turns ${\displaystyle T_{x}M}$ into a vector space.

Equivalence of the definitions

For ${\displaystyle x\in M}$ and a differentiable curve ${\displaystyle \gamma$ :(-1,1)\to M} such that ${\displaystyle \gamma (0)=x,}$ define ${\displaystyle {D_{\gamma }}(f):=(f\circ \gamma )'(0)}$ (where the derivative is taken in the ordinary sense because ${\displaystyle f\circ \gamma }$ is a function from ${\displaystyle (-1,1)}$ to ${\displaystyle \mathbb {R} }$). One can ascertain that ${\displaystyle D_{\gamma }(f)}$ is a derivation at the point ${\displaystyle x,}$ and that equivalent curves yield the same derivation. Thus, for an equivalence class ${\displaystyle \gamma '(0),}$ we can define ${\displaystyle {D_{\gamma '(0)}}(f):=(f\circ \gamma )'(0),}$ where the curve ${\displaystyle \gamma \in \gamma '(0)}$ has been chosen arbitrarily. The map ${\displaystyle \gamma '(0)\mapsto D_{\gamma '(0)}}$ is a vector space isomorphism between the space of the equivalence classes ${\displaystyle \gamma '(0)}$ and that of the derivations at the point ${\displaystyle x.}$

Definition via cotangent spaces

Again, we start with a ${\displaystyle C^{\infty }}$ manifold ${\displaystyle M}$ and a point ${\displaystyle x\in M}$. Consider the ideal ${\displaystyle I}$ of ${\displaystyle C^{\infty }(M)}$ that consists of all smooth functions ${\displaystyle f}$ vanishing at ${\displaystyle x}$, i.e., ${\displaystyle f(x)=0}$. Then ${\displaystyle I}$ and ${\displaystyle I^{2}}$ are both real vector spaces, and the quotient space ${\displaystyle I/I^{2}}$ can be shown to be isomorphic to the cotangent space ${\displaystyle T_{x}^{*}M}$ through the use of Taylor's theorem. The tangent space ${\displaystyle T_{x}M}$ may then be defined as the dual space of ${\displaystyle I/I^{2}}$.

While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry.

If ${\displaystyle D}$ is a derivation at ${\displaystyle x}$, then ${\displaystyle D(f)=0}$ for every ${\displaystyle f\in I^{2}}$, which means that ${\displaystyle D}$ gives rise to a linear map ${\displaystyle I/I^{2}\to \mathbb {R} }$. Conversely, if ${\displaystyle r:I/I^{2}\to \mathbb {R} }$ is a linear map, then ${\displaystyle D(f):=r\left((f-f(x))+I^{2}\right)}$ defines a derivation at ${\displaystyle x}$. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.

Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector ${\displaystyle v}$ in ${\displaystyle \mathbb {R} ^{n}}$, one defines the corresponding directional derivative at a point ${\displaystyle x\in \mathbb {R} ^{n}}$ by

${\displaystyle \forall f\in {C^{\infty }}(\mathbb {R} ^{n}):\qquad (D_{v}f)(x):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[f(x+tv)]\right|_{t=0}=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(x).}$

This map is naturally a derivation at ${\displaystyle x}$. Furthermore, every derivation at a point in ${\displaystyle \mathbb {R} ^{n}}$ is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if ${\displaystyle v}$ is a tangent vector to ${\displaystyle M}$ at a point ${\displaystyle x}$ (thought of as a derivation), then define the directional derivative ${\displaystyle D_{v}}$ in the direction ${\displaystyle v}$ by

${\displaystyle \forall f\in {C^{\infty }}(M):\qquad {D_{v}}(f):=v(f).}$

If we think of ${\displaystyle v}$ as the initial velocity of a differentiable curve ${\displaystyle \gamma }$ initialized at ${\displaystyle x}$, i.e., ${\displaystyle v=\gamma '(0)}$, then instead, define ${\displaystyle D_{v}}$ by

${\displaystyle \forall f\in {C^{\infty }}(M):\qquad {D_{v}}(f):=(f\circ \gamma )'(0).}$

Basis of the tangent space at a point

For a ${\displaystyle C^{\infty }}$ manifold ${\displaystyle M}$, if a chart ${\displaystyle \varphi =(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}}$ is given with ${\displaystyle p\in U}$, then one can define an ordered basis ${\textstyle \left\{\left.{\frac {\partial }{\partial x^{1}}}\right|_{p},\dots ,\left.{\frac {\partial }{\partial x^{n}}}\right|_{p}\right\}}$ of ${\displaystyle T_{p}M}$ by

${\displaystyle \forall i\in \{1,\ldots ,n\},~\forall f\in {C^{\infty }}(M):\qquad {\left.{\frac {\partial }{\partial x^{i}}}\right|_{p}}(f):=\left({\frac {\partial }{\partial x^{i}}}{\Big (}f\circ \varphi ^{-1}{\Big )}\right){\Big (}\varphi (p){\Big )}.}$

Then for every tangent vector ${\displaystyle v\in T_{p}M}$, one has

${\displaystyle v=\sum _{i=1}^{n}v^{i}\left.{\frac {\partial }{\partial x^{i}}}\right|_{p}.}$

This formula therefore expresses ${\displaystyle v}$ as a linear combination of the basis tangent vectors ${\textstyle \left.{\frac {\partial }{\partial x^{i}}}\right|_{p}\in T_{p}M}$ defined by the coordinate chart ${\displaystyle \varphi :U\to \mathbb {R} ^{n}}$.^[4]

The derivative of a map

Every smooth (or differentiable) map ${\displaystyle \varphi :M\to N}$ between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces:

${\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to T_{\varphi (x)}N.}$

If the tangent space is defined via differentiable curves, then this map is defined by

${\displaystyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):=(\varphi \circ \gamma )'(0).}$

If, instead, the tangent space is defined via derivations, then this map is defined by

${\displaystyle [\mathrm {d} {\varphi }_{x}(D)](f):=D(f\circ \varphi ).}$

The linear map ${\displaystyle \mathrm {d} {\varphi }_{x}}$ is called variously the derivative, total derivative, differential, or pushforward of ${\displaystyle \varphi }$ at ${\displaystyle x}$. It is frequently expressed using a variety of other notations:

${\displaystyle D\varphi _{x},\qquad (\varphi _{*})_{x},\qquad \varphi '(x).}$

In a sense, the derivative is the best linear approximation to ${\displaystyle \varphi }$ near ${\displaystyle x}$. Note that when ${\displaystyle N=\mathbb {R} }$, then the map ${\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} }$ coincides with the usual notion of the differential of the function ${\displaystyle \varphi }$. In local coordinates the derivative of ${\displaystyle \varphi }$ is given by the Jacobian.

An important result regarding the derivative map is the following:

This is a generalization of the inverse function theorem to maps between manifolds.