Exponential mapping
A mapping of the tangent space of a manifold into
. It is defined by a connection given on
and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
1) Let be a
-manifold with an affine connection, let
be a point in
, let
be the tangent space to
at
, let
be a non-zero vector in
, and let
be the geodesic passing through
in the direction of
. There is an open neighbourhood
of the point
in
and an open neighbourhood
of
in
such that the mapping
is a diffeomorphism of
onto
. This mapping is called the exponential mapping at
and is denoted by
. A neighbourhood
is called normal if: 1) the mapping
maps
onto
diffeomorphically; and 2)
and
imply that
. In this case
is said to be a normal neighbourhood of the point
in the manifold
. Every
has a convex normal neighbourhood
: Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in
. If
is a complete Riemannian manifold, then
is a surjective mapping of
onto
.
2) Let be a Lie group with identity
and let
be the corresponding Lie algebra consisting of the tangent vectors to
at
. For every vector
there is a unique differentiable homomorphism
of the group
into
such that the tangent vector to
at
coincides with
. The mapping
is called the exponential mapping of the algebra
into the group
. There is an open neighbourhood
of the point
in
and an open neighbourhood
of
in
such that
is a diffeomorphism of
onto
. Let
be some basis for the algebra
. The mapping
is a coordinate system on
; these coordinates are called canonical.
The concept of an exponential mapping of a Lie group can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on
that are invariant relative to the group of left translations and the set of bilinear functions
. It turns out that the exponential mapping
of the algebra
into the group
coincides with the mapping
of the tangent space of
into the manifold
at the point
in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function
.
References
[1] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
Exponential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exponential_mapping&oldid=12230