Search results
Results from the WOW.Com Content Network
If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N. In differential geometry , pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.
This projection maps each element of the tangent space to the single point . The tangent bundle comes equipped with a natural topology (described in a section below ). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces ).
A differentiable map f : M → N is said to have constant rank if the rank of f is the same for all p in M. Constant rank maps have a number of nice properties and are an important concept in differential topology. Three special cases of constant rank maps occur. A constant rank map f : M → N is
is an injective function at every point p of M (where T p X denotes the tangent space of a manifold X at a point p in X and D p f is the derivative (pushforward) of the map f at point p). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M: [2] = .
The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. [2]
These two bits of data, a direction and a magnitude, thus determine a tangent vector at the base point. The map from tangent vectors to endpoints smoothly sweeps out a neighbourhood of the base point and defines what is called the exponential map, defining a local coordinate chart at that base point. The neighbourhood swept out has similar ...
As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold. Let F(U) be the set of all sections on U. F(U) always contains at least one element, namely the zero section: the function s that maps every element x of U to the zero element of the vector space π −1 ({x}).
The differential assigns to each point a linear map from the tangent space to the real numbers. In this case, each tangent space is naturally identifiable with the real number line, and the linear map R → R {\displaystyle \mathbb {R} \to \mathbb {R} } in question is given by scaling by f ′ ( x 0 ) . {\displaystyle f'(x_{0}).}