Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometryexponential mapsRiemannian exponential mapRiemannian exponential mapping
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.wikipedia
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Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.
A Riemannian manifold M is geodesically complete if for all p ∈ M, the exponential map exp p is defined for all v ∈ T p M, i.e. if any geodesic γ(t) starting from p is defined for all values of the parameter t ∈ R. The Hopf–Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space.

Tangent space

tangent planetangenttangent vector
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
However even if exp p is defined on the whole tangent space, it will in general not be a global diffeomorphism.
Over a compact subset of M, this follows by fixing a Riemannian metric on M and using the exponential map for that metric.

Affine connection

connectionaffineaffine connections
. An affine connection on
This follows from the Picard–Lindelöf theorem, and allows for the definition of an exponential map associated to the affine connection.

Geodesic

geodesicsgeodesic flowgeodesic equation
. Then there is a unique geodesic
Thus, G^t(V) = exp(tV) is the exponential map of the vector tV.

Glossary of Riemannian and metric geometry

injectivity radiusgeodesic metric spaceproper
The radius of the largest ball about the origin in T p M that can be mapped diffeomorphically via exp p is called the injectivity radius of M at p.
Exponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)

Hopf–Rinow theorem

completecomplete manifoldgeodesically complete
The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space (which justifies the usual term geodesically complete for a manifold having an exponential map with this property).

Normal coordinates

geodesic normal coordinatesnormal coordinate systemnormal ball
This motivates the definition of geodesic normal coordinates on a Riemannian manifold.
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 tangent space at p.

Cut locus (Riemannian manifold)

cut locusRiemannian cut
The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
It is a standard result that for sufficiently small v in T_p M, the curve defined by the Riemannian exponential map, for t belonging to the interval [0,1] is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints.

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration.
It is the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ p (in other words, the image of σ p under the exponential map at p).

Gauss's lemma (Riemannian geometry)

Gauss's lemmalemma of Gauss
An important property of the exponential map is the following lemma of Gauss (yet another Gauss's lemma): given any tangent vector v in the domain of definition of exp p, and another vector w based at the tip of v (hence w is actually in the double-tangent space T v (T p M)) and orthogonal to v, remains orthogonal to v when pushed forward via the exponential map.
The exponential map is a mapping from the tangent space at p to M:

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration.
It is the Gauss curvature of the \sigma-section at p; here \sigma-section is a locally defined piece of surface which has the plane \sigma as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of \sigma under the exponential map at p.

Exponential map (Lie theory)

exponential mapexponential mappingexponential coordinates
In the case of Lie groups with a bi-invariant metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of the Lie group.
If G is compact, it has a Riemannian metric invariant under left and right translations, and the Lie-theoretic exponential map for G coincides with the exponential map of this Riemannian metric.

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
be a differentiable manifold and

Line (geometry)

linestraight linelines
allows one to define the notion of a straight line through the point

Tangent vector

tangent vectorstangent directionstangent
be a tangent vector to the manifold at

Picard–Lindelöf theorem

existence and uniquenessCauchy-Lipschitz theoremCauchy–Lipschitz theorem
This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature.

Ordinary differential equation

ordinary differential equationsordinaryODE
This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.

Metric space

metricmetric spacesmetric geometry
The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space (which justifies the usual term geodesically complete for a manifold having an exponential map with this property).

Compact space

compactcompact setcompactness
In particular, compact manifolds are geodesically complete.

Identity function

identity mapidentity operatoridentity
However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of T p M on which the exponential map is an embedding (i.e., the exponential map is a local diffeomorphism).