Definition:Tangent Space

From ProofWiki
Jump to navigation Jump to search


This page has been identified as a candidate for refactoring of medium complexity.
In particular: Pay particular attention to the coherence of the presentation and references
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


Definition

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


This article is incomplete.
In particular: Link to Definition:Tangent Vector/Definition 2
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Stub}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


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


That is, it is the set of all tangent vectors to $M$ 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

  • Results about tangent spaces can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Tangent_Space&oldid=743152"