# Tangent bundle

of a differentiable manifold $M$

The vector bundle $\tau:TM\rightarrow M$, also denoted $\tau (M)$, whose total space $TM$ is given by the union of the tangent spaces $TM|_x$ to $M$ at $x\in M$, consisting of the tangent vectors to $M$, and with projection $\tau$ mapping $TM|_x$ to the point $x$. A section of the tangent bundle $\tau (M)$ is a vector field on $M$ (cf. Vector field on a manifold). An atlas on the manifold $TM$ is defined through an atlas of the manifold $M$. The bundle $\tau (M)$ is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.

Associated with the tangent bundle is the frame bundle of the manifold $M$, which is a principal bundle. The bundle $\tau^*(M)$ dual to the tangent bundle $\tau (M)$, called the cotangent bundle, consists of the cotangent spaces to $M$. Its sections are the differential or Pfaffian forms (cf. Pfaffian form).

A differentiable mapping $h:M\rightarrow N$ induces a morphism of tangent bundles $\tau(M)\rightarrow \tau(N)$; the corresponding mapping of the total spaces $Th:TM\rightarrow TN$ is called the tangent mapping to $h$ (or differential of $h$). In particular, if $i:M\rightarrow N$ is an immersion (cf. Immersion of a manifold), then $\tau(M)$ is a subbundle of the induced vector bundle $i^*\tau(N)$. The quotient bundle $\nu (i)=i^*\tau(N)/\tau(M)$ is called the normal bundle of the immersion. Dually, if $j:M\rightarrow N$ is a submersion, then the quotient bundle $\tau(M)/j^*\tau(N)$ is called a subbundle of $j$. If one chooses for $M$ and $N$, $TM$ and $M$ respectively, and $h=\tau:TM\rightarrow M$, then $\tau^*\tau (M)$ is called the tangent bundle of second order.

If $\tau (M)$ is trivial, then $M$ is called a parallelizable manifold.

How to Cite This Entry:
Tangent bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_bundle&oldid=41519
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article