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.
In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...
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 texture map (left). The corresponding normal map in tangent space (center). The normal map applied to a sphere in object space (right). Normal map reuse is made possible by encoding maps in tangent space. The tangent space is a vector space, which is tangent to the model's surface. The coordinate system varies smoothly (based on the ...
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]
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
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .