In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, Gauss's lemma asserts that any sufficiently small
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
centered at a point in a
Riemannian manifold is perpendicular to every
geodesic through the point. More formally, let ''M'' be a
Riemannian manifold, equipped with its
Levi-Civita connection, and ''p'' a point of ''M''. The
exponential map is a mapping from the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at ''p'' to ''M'':
:
which is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
in a neighborhood of zero. Gauss' lemma asserts that the image of a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
of sufficiently small radius in ''T''
p''M'' under the exponential map is perpendicular to all
geodesics originating at ''p''. The lemma allows the exponential map to be understood as a radial
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
, and is of fundamental importance in the study of geodesic
convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
and
normal coordinates
In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tang ...
.
Introduction
We define the exponential map at
by
:
where
is the unique
geodesic with
and tangent
and
is chosen small enough so that for every
the geodesic
is defined. So, if
is complete, then, by the
Hopf–Rinow theorem
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem ...
,
is defined on the whole tangent space.
Let
be a curve differentiable in
such that
and
. Since
, it is clear that we can choose
. In this case, by the definition of the differential of the exponential in
applied over
, we obtain:
:
So (with the right identification
) the differential of
is the identity. By the implicit function theorem,
is a diffeomorphism on a neighborhood of
. The Gauss Lemma now tells that
is also a radial isometry.
The exponential map is a radial isometry
Let
. In what follows, we make the identification
.
Gauss's Lemma states:
Let
and
. Then,
For
, this lemma means that
is a radial isometry in the following sense: let
, i.e. such that
is well defined.
And let
. Then the exponential
remains an isometry in
, and, more generally, all along the geodesic
(in so far as
is well defined)! Then, radially, in all the directions permitted by the domain of definition of
, it remains an isometry.
Proof
Recall that
:
We proceed in three steps:
* ''
'' : let us construct a curve
such that
and
. Since
, we can put
.
Therefore,
where
is the parallel transport operator and
. The last equality is true because
is a geodesic, therefore
is parallel.
Now let us calculate the scalar product
.
We separate
into a component
parallel to
and a component
normal to
. In particular, we put
,
.
The preceding step implies directly:
:
::
We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have:
* ''
'' :
Let us define the curve
:
Note that
:
Let us put:
:
and we calculate:
:
and
:
Hence
:
We can now verify that this scalar product is actually independent of the variable
, and therefore that, for example:
:
because, according to what has been given above:
:
being given that the differential is a linear map. This will therefore prove the lemma.
* We verify that ''
'': this is a direct calculation. Since the maps
are geodesics,
:
Since the maps
are geodesics,
the function
is constant. Thus,
:
See also
*
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
*
Metric tensor
References
*
{{Manifolds
Articles containing proofs
Lemmas
Riemannian geometry
Riemannian manifolds
Theorems in Riemannian geometry