In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to...

Important special cases include: exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra...

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...

Euclidean geometry Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry) Finsler metric A generalization of Riemannian manifolds...

function's current value Exponential map (Riemannian geometry), in Riemannian geometry Exponential map (Lie theory), in Lie theory Exponential notation, also known...

integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation...

flow Exponential map (Lie theory) Exponential map (Riemannian geometry) Injectivity radius Geodesic deviation equation Jacobi field Riemannian symmetric...

differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric...

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...

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated...

there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant...

numbers. The conformal maps are described by linear fractional transformations in each case. In Riemannian geometry, two Riemannian metrics g {\displaystyle...

algebraic geometry Normal coordinates, in differential geometry, local coordinates obtained from the exponential map (Riemannian geometry) Normal distribution...

inverse of the exponential map at p is a polar coordinate system. Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial...

symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically...

The exponential map exp p : T p ( S ) ⟶ S {\displaystyle \exp _{p}:T_{p}(S)\longrightarrow S} is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard...

is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M. On a (pseudo-)Riemannian manifold...

singly exponential in the number of the variables. A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has...

v} in T p M {\displaystyle T_{p}M} , the curve defined by the Riemannian exponential map, γ ( t ) = exp p ⁡ ( t v ) {\displaystyle \gamma (t)=\exp _{p}(tv)}...

⁠, then the exponential map takes the Lie algebra of G {\displaystyle G} into ⁠ G {\displaystyle G} ⁠; thus, we have an exponential map for all matrix...

Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature...

new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the...

Introduction to Riemannian Manifolds. Graduate Texts in Mathematics. Springer International Publishing AG. O'Neill, Barrett (1983). Semi-Riemannian Geometry. Academic...

classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction...

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends...

means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms...

is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the...

which is a complete Riemannian metric on D with curvature −1 (isometric to the hyperbolic plane). Namely: every holomorphic map from D to itself is distance-decreasing...

a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. Let V be a smooth vector field on a smooth...

Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity...