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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.
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] = .
A real-valued function : is said to belong to () if and only if for every coordinate chart :, the map : [] is infinitely differentiable. Note that C ∞ ( M ) {\displaystyle {C^{\infty }}(M)} is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.
The tangent bundle of the circle is also trivial and isomorphic to . Geometrically, this is a cylinder of infinite height. The only tangent bundles that can be readily visualized are those of the real line and the unit circle , both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to ...
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]
It is also denoted by Tf and called the tangent map. At each point of M, this is a linear transformation from one tangent space to another: (): (). The rank of f at p is the rank of this linear transformation. Usually the rank of a function is a pointwise property.
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
Smooth maps between manifolds induce linear maps between tangent spaces: for :, at each point the pushforward (or differential) maps tangent vectors at to tangent vectors at (): ,: (), and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism: :.