Exponential mapping

A mapping of the tangent space of a manifold $ M $ into $ M $. It is defined by a connection given on $ M $ and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.

1) Let $ M $ be a $ C ^ \infty $- manifold with an affine connection, let $ p $ be a point in $ M $, let $ M _ {p} $ be the tangent space to $ M $ at $ p $, let $ X $ be a non-zero vector in $ M _ {p} $, and let $ t \rightarrow \gamma _ {X} ( t) $ be the geodesic passing through $ p $ in the direction of $ X $. There is an open neighbourhood $ N _ {0} $ of the point $ 0 $ in $ M _ {p} $ and an open neighbourhood $ N _ {p} $ of $ p $ in $ M $ such that the mapping $ X \rightarrow \gamma _ {X} ( 1) $ is a diffeomorphism of $ N _ {0} $ onto $ N _ {p} $. This mapping is called the exponential mapping at $ p $ and is denoted by $ \mathop{\rm exp} $. A neighbourhood $ N _ {0} $ is called normal if: 1) the mapping $ \mathop{\rm exp} $ maps $ N _ {0} $ onto $ N _ {p} $ diffeomorphically; and 2) $ X \in N _ {0} $ and $ 0 \leq t \leq 1 $ imply that $ t X \in N _ {0} $. In this case $ N _ {p} $ is said to be a normal neighbourhood of the point $ p $ in the manifold $ M $. Every $ p \in M $ has a convex normal neighbourhood $ N _ {p} $: Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in $ N _ {p} $. If $ M $ is a complete Riemannian manifold, then $ \mathop{\rm exp} $ is a surjective mapping of $ M _ {p} $ onto $ M $.

2) Let $ G $ be a Lie group with identity $ e $ and let $ \mathfrak g $ be the corresponding Lie algebra consisting of the tangent vectors to $ G $ at $ e $. For every vector $ X \in \mathfrak g $ there is a unique differentiable homomorphism $ \theta $ of the group $ \mathbf R $ into $ G $ such that the tangent vector to $ \theta ( \mathbf R ) $ at $ e $ coincides with $ X $. The mapping $ X \rightarrow \mathop{\rm exp} X = \theta ( 1) $ is called the exponential mapping of the algebra $ \mathfrak g $ into the group $ G $. There is an open neighbourhood $ N _ {0} $ of the point $ 0 $ in $ \mathfrak g $ and an open neighbourhood $ N _ {e} $ of $ e $ in $ G $ such that $ \mathop{\rm exp} $ is a diffeomorphism of $ N _ {0} $ onto $ N _ {e} $. Let $ X _ {1} \dots X _ {n} $ be some basis for the algebra $ \mathfrak g $. The mapping $ \mathop{\rm exp} ( x _ {1} X _ {1} + {} \dots + x _ {n} X _ {n} ) \rightarrow ( x _ {1} \dots x _ {n} ) $ is a coordinate system on $ N _ {e} $; these coordinates are called canonical.

The concept of an exponential mapping of a Lie group $ G $ can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on $ G $ that are invariant relative to the group of left translations and the set of bilinear functions $ \alpha : \mathfrak g \times \mathfrak g \rightarrow \mathfrak g $. It turns out that the exponential mapping $ \mathop{\rm exp} $ of the algebra $ \mathfrak g $ into the group $ G $ coincides with the mapping $ \mathop{\rm exp} $ of the tangent space of $ \mathfrak g $ into the manifold $ G $ at the point $ e $ in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function $ \alpha $.

References

[1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)