In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, Gauss's lemma asserts that any sufficiently small
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
centered at a point in a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
is perpendicular to every
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
through the point. More formally, let ''M'' be a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, equipped with its
Levi-Civita connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
, and ''p'' a point of ''M''. The
exponential map is a mapping from the
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
at ''p'' to ''M'':
:
which is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
in a neighborhood of zero. Gauss' lemma asserts that the image of a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of sufficiently small radius in ''T''
p''M'' under the exponential map is perpendicular to all
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s 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 and
normal coordinates
In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a torsion tensor, symmetric affine connection are a local coordinate system in a neighborhood (mathematics), neighborhood of ''p'' obtained by ...
.
Introduction
We define the exponential map at
by
:
where
is the unique
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
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,
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, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
*
Metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
References
*
{{Manifolds
Articles containing proofs
Lemmas
Riemannian geometry
Riemannian manifolds
Theorems in Riemannian geometry