The exponential is the exponential map, in this case the matrix exponential defined by putting the matrix into the usual power series for the exponential function.
32.
The exponential map is known not to be surjective in this case, even though it is surjective on the whole group " L ".
33.
Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is surjective ( onto ).
34.
With some additional abuse of language, one notes that the exponential map provides a map from vectors in a tangent space to points in an underlying manifold.
35.
This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors.
36.
The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
37.
In particular, cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
38.
In particular, cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
39.
In particular the exponential map is a polynomial mapping of \ mathfrak { n } onto " N ", with polynomial inverse given by the logarithm.
40.
A principal homogeneous space of is a manifold abstractly characterized by having a exponential map of the Lie algebra and in this way obtain, locally, a group action on.
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