Suppose and are smooth manifolds with tangent bundles and , and suppose is a smooth mapping. Then the tangent map of is the map defined as follows: If for some , then we can represent by some curve with and . Now is defined as the tangent vector in represented by the curve . Thus, since , it follows that .
Note that if is a mapping as in the definition, then the tangent map is a mapping
whereas the pullback (http://planetmath.org/PullbackOfAKForm) of is a mapping
For this reason, the tangent map is also sometimes called the pushforward map. That is, a pullback takes objects from to , and a pushforward takes objects from to .
Sometimes, the tangent map of is also denoted by . However, the motivation for denoting the tangent map by is that if and are open subsets in and , then is simply the Jacobian of .
|Date of creation||2013-03-22 14:06:19|
|Last modified on||2013-03-22 14:06:19|
|Last modified by||matte (1858)|
|Synonym||push forward map|