The theorem holds also for Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map (; ).
42.
Conversely, the surjectivity of the exponential map, together with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices.
43.
Take for example the complex plane under the exponential map : the image is "'C "'- { 0 }, which clearly is not simply connected.
44.
More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices ( in the more elementary sense of Lattice ( group ) ) in the Lie algebra.
45.
On the open ball of this radius, the exponential map at " p " is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius.
46.
The exponential map from the Lie algebra to the Lie group is not always " C " " Fr�chet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.
47.
In general, the exponential map is only " locally defined ", that is, it only takes a small neighborhood of the origin at, to a neighborhood of in the manifold.
48.
The exponential map is one-to-one in a neighborhood of the identity in, hence the composition, where is the Lie algebra isomorphism, is onto an open neighborhood containing the identity.
49.
This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish a tight connection between properties of a modular Lie algebra and the corresponding algebraic group.
50.
Putting this into the Taylor series for the exponential map and using the property "'e 12 "'2 = " 1 results in a bivector version of Euler's formula,
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