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 setting for studying many of the concepts of mathematical analysis and geometry.
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 connection. It is a generalization of the notion of a "straight line".
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 -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.
In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold 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.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.
In mathematical physics, 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 requirement of positive-definiteness is relaxed.
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space.
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
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 to the context of certain types of metric spaces.
In mathematics, a Killing vector field, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of
In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces . The concept captures the limiting behavior of finite configurations in the spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.
In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete, this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.
