tangent map

Definition 1.

Suppose X and Y are smooth manifoldsMathworldPlanetmath with tangent bundles TX and TY, and suppose f:XY is a smooth mapping. Then the tangent map of f is the map Df:TXTY defined as follows: If vTx(X) for some xX, then we can represent v by some curve c:IX with c(0)=x and I=(-1,1). Now (Df)(v) is defined as the tangent vector in T(Y) represented by the curve fc:IY. Thus, since (fc)(0)=f(x), it follows that (Df)(v)Tf(x)(Y).


Suppose X and Y are a smooth manifolds.

  • If idX is the identity mapping on X, then DidX is the identity mapping on TX.

  • Suppose X,Y,Z are smooth manifolds, and f,g are mappings f:XY, g:YZ. Then

  • If f:XY is a diffeomorphism, then the inverse of Df is a diffeomorphism, and



Note that if f:XY is a mapping as in the definition, then the tangent map is a mapping


whereas the pullback (http://planetmath.org/PullbackOfAKForm) of f is a mapping


For this reason, the tangent map is also sometimes called the pushforward map. That is, a pullback takes objects from Y to X, and a pushforward takes objects from X to Y.

Sometimes, the tangent map of f is also denoted by f. However, the motivation for denoting the tangent map by Df is that if X and Y are open subsets in n and m, then Df is simply the Jacobian of f.

Title tangent map
Canonical name TangentMap
Date of creation 2013-03-22 14:06:19
Last modified on 2013-03-22 14:06:19
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Definition
Classification msc 53-00
Synonym push forward map
Synonym pushforward
Synonym pushforward map
Related topic PullbackOfAKForm
Related topic FlowBoxTheorem