HOME

TheInfoList



OR:

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 ...
, an exponential map is a map from a subset of a
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 ...
T''p''''M'' of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(or
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
) ''M'' to ''M'' itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.


Definition

Let be a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
and a point of . An affine connection on allows one to define the notion of a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
through the point .A source for this section is , which uses the term "linear connection" where we use "affine connection" instead. Let be a
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
to the manifold at . Then there is a unique
geodesic In geometry, a geodesic () is a curve representing in some sense the 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 connecti ...
satisfying with initial tangent vector . The corresponding exponential map is defined by . In general, the exponential map is only ''locally defined'', that is, it only takes a small neighborhood of the origin at , to a neighborhood of in the manifold. This is because it relies on the theorem of existence and uniqueness for
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
.


Properties

Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and goes in that direction, for a unit time. Since ''v'' corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define exp''p''(''v'') = β(, ''v'', ) where β is the unit-speed geodesic (geodesic parameterized by arc length) going in the direction of ''v''. As we vary the tangent vector ''v'' we will get, when applying exp''p'', different points on ''M'' which are within some distance from the base point ''p''—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold. 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 ...
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 In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
(which justifies the usual term geodesically complete for a manifold having an exponential map with this property). In particular,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
manifolds are geodesically complete. However even if exp''p'' is defined on the whole tangent space, it will in general not be a global
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 ...
. However, its differential at the origin of the tangent space is the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
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). 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 This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or prov ...
of ''M'' at ''p''. 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. 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'', ''w'' remains orthogonal to ''v'' when pushed forward via the exponential map. This means, in particular, that the boundary sphere of a small ball about the origin in T''p''''M'' is orthogonal to the geodesics in ''M'' determined by those vectors (i.e., the geodesics are ''radial''). This motivates the definition of
geodesic 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 tange ...
on 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 In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a p ...
is intuitively defined as the
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...
of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point ''p'' in consideration. Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through ''p'' determined by the image under exp''p'' of a 2-dimensional subspace of T''p''''M''.


Relationships to exponential maps in Lie theory

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. In general, Lie groups do not have a bi-invariant metric, though all connected semi-simple (or reductive) Lie groups do. The existence of a bi-invariant ''Riemannian'' metric is stronger than that of a pseudo-Riemannian metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact (or abelian) Lie group has such a Riemannian metric. Take the example that gives the "honest" exponential map. Consider the
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
R+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point ''y'', we introduce the modified inner product \langle u,v\rangle_y = \frac multiplying them as usual real numbers but scaling by ''y''2 (this is what makes the metric left-invariant, for left multiplication by a factor will just pull out of the inner product, twice — canceling the square in the denominator). Consider the point 1 ∈ R+, and ''x'' ∈ R an element of the tangent space at 1. The usual straight line emanating from 1, namely ''y''(''t'') = 1 + ''xt'' covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed ("constant speed", remember, is not going to be the ordinary constant speed, because we're using this funny metric). To do this we reparametrize by arc length (the integral of the length of the tangent vector in the norm , \cdot, _y induced by the modified metric): s(t) = \int_0^t , x, _ d\tau = \int_0^t \frac d\tau = , x, \int_0^t \frac = \frac \ln, 1 + tx, and after inverting the function to obtain as a function of , we substitute and get y(s) = e^ Now using the unit speed definition, we have \exp_1(x) = y(, x, _1) = y(, x, ), giving the expected ''e''''x''. The Riemannian distance defined by this is simply \operatorname(a, b) = \left, \ln\left(\frac b a\right)\.


See also

* List of exponential topics


Notes


References

* . See Chapter 1, Sections 2 and 3. * . See Chapter 3. * * . * . {{DEFAULTSORT:Exponential Map Differential geometry Riemannian geometry