exponential map in a sentence

Examples

1. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
2. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.
3. Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.
4. The exponential map provides a diffeomorphism between a neighborhood of the origin in the and a neighborhood of the identity in the.
5. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the exponential map.
6. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.
7. In other words, the Rodrigues'formula provides an algorithm to compute the exponential map from to without computing the full matrix exponential.
8. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.
9. Bivectors are related to rotors through the exponential map ( which applied to bivectors generates rotors and rotations using De Moivre's formula ).
10. The universal cover group above can be constructed as the image of the Lie algebra under the compact, the exponential map need not be surjective.