Definition:Tangent Space

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Real Submanifold

Let $M$ be a real submanifold of $\R^n$ of dimension $d$.

Let $p \in M$.

Using Local Submersions

Let $U$ be a open neighborhood of $p$ in $\R^n$ and $\phi : U \to \R^{n - d}$ be a submersion such that:

$M \cap U = \phi^{-1} \sqbrk 0$.

The tangent space of $M$ at $p$ is:

$T_p M = \ker \d \map \phi p$

where $\d \map \phi p$ is the differential of $\phi$ at $p$.

Using Local Embdeddings

Let $U$ be a open set of $\R^d$ and $\phi: U \to \R^n$ be an embedding such that:

$p \in \phi \sqbrk U \subset M$

The tangent space of $M$ at $p$ is:

$T_p M = \Img {\map {\paren {\d \phi} } {\map {\phi^{-1} } p} }$

where $\map {\paren {\d \phi} } {\map {\phi^{-1} } p}$ is the differential of $\phi$ at $\map {\phi^{-1} } p$.

Using Local Immersions

Geometric Tangent Space

Let $a \in \R^n$ be an element of the $n$-dimensional Euclidean space.

The geometric tangent space to $\R^n$ at $a$, denoted by $\R^n_a$, is the cartesian product of the singleton $\set a$ and $\R^n$:

$\R^n_a := \set a \times \R^n = \set {\tuple {a, v} : v \in \R^n}$

Differentiable Manifold

There are various ways to construct the tangent space of a differentiable manifold.

Let $M$ be a smooth manifold of dimension $m$.

Let $p \in M$.

Abstract Definition

Let $A$ be an atlas of $M$.

Let $B = \set {\struct {U, \phi} \in A: p \in U}$.

The tangent space of $M$ at $p$ is the real vector space

$\ds \paren {\coprod_{b \mathop \in B} \set b \times \R^m} / \sim$

where $\sim$ is the equivalence relation defined by:

$\tuple {i, \xi} \sim \tuple {j, \eta} \iff \d \map {\map {\paren {\phi_j \circ \phi_i^{-1} } } {\map {\phi_i} p} } \xi = \eta$

where $\d \map {\map {\paren {\phi_j \circ \phi_i^{-1} } } {\map {\phi_i} p} } \xi$ is the differential of $\phi_j \circ \phi_i^{-1}$ at $\map {\phi_i} p$.

Addition is defined by:

$\eqclass {\tuple {i, \xi} } \sim + \eqclass {\tuple {j, \eta} } \sim = \eqclass {\tuple {i, \d \map {\map {\paren {\phi_j \circ \phi_i^{-1} } } {\map {\phi_i} p} } \xi + \eta} } \sim$

and scalar multiplication by:

$\lambda \cdot \eqclass {\tuple {i, \xi} } \sim = \eqclass {\tuple {i, \lambda \cdot \xi} } \sim$

As a space of germs of curves

As a space of derivations

Let $M$ be a smooth manifold.

Let $p \in M$ be a point in $M$.

Let $v: \map {C^\infty} M \to \R$ be a derivation at $p$.

The tangent space at $p$, denoted by $T_p M$, is the set of all derivations of $\map {\CC^\infty} M$ at $p$.

Equivalence of Definitions

While the above constructions are not the same, there are very closely related. See Equivalence of Definitions of Tangent Space.

Also see