The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below.
Affine connection Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2).
is called bi-Lipschitz if there are positive constants c and C such that for any x and y in X Boundary at infinity.
In general, a construction that may be regarded as a space of directions at infinity.
Cartan connection Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.
Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to Rn via the exponential map; for metric spaces, the statement that a connected, simply connected complete geodesic metric space with non-positive curvature in the sense of Alexandrov is a (globally) CAT(0) space.
of a Riemannian manifold is the supremum of radii of balls centered at
[4] Cotangent bundle Covariant derivative Cubical complex Cut locus Diameter of a metric space is the supremum of distances between pairs of points.
Ehresmann connection Einstein manifold Euclidean geometry Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry) Finsler metric A generalization of Riemannian manifolds where the scalar product on the tangent space is replaced by a norm.
First fundamental form for an embedding or immersion is the pullback of the metric tensor.
Flat manifold Geodesic is a curve which locally minimizes distance.
For complete manifolds, if the injectivity radius at p is a finite number r, then either there is a geodesic of length 2r which starts and ends at p or there is a point q conjugate to p (see conjugate point above) and on the distance r from p.[6] For a closed Riemannian manifold the injectivity radius is either half the minimal length of a closed geodesic or the minimal distance between conjugate points on a geodesic.
Infranilmanifold Given a simply connected nilpotent Lie group N acting on itself by left multiplication and a finite group of automorphisms F of N one can define an action of the semidirect product
measures "how efficiently rectifiable loops are coarsely contractible with respect to their length".
Length space Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.
[10][11] Mean curvature Metric ball Metric tensor Minkowski space Minimal surface is a submanifold with (vector of) mean curvature zero.
, compact hyperbolic manifolds are classified by their fundamental group.
Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented
It also can be defined as a factor of a connected nilpotent Lie group by a lattice.
Normal bundle: associated to an embedding of a manifold M into an ambient Euclidean space
[14] Pseudo-Riemannian manifold Quasi-convex subspace of a metric space
For example, any map between compact metric spaces is a quasi isometry.
[16] Ray is a one side infinite geodesic which is minimizing on each interval.
[17] Real tree Rectifiable curve Ricci curvature Riemann The mathematician after whom Riemannian geometry is named.
Scalar curvature Second fundamental form is a quadratic form on the tangent space of hypersurface, usually denoted by II, an equivalent way to describe the shape operator of a hypersurface, It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space.
If n is a unit normal field to M and v is a tangent vector then (there is no standard agreement whether to use + or − in the definition).
Spherical geometry Submetry A short map f between metric spaces is called a submetry[18] if there exists R > 0 such that for any point x and radius r < R the image of metric r-ball is an r-ball, i.e.
Sub-Riemannian manifold Symmetric space Riemannian symmetric spaces are Riemannian manifolds in which the geodesic reflection at any point is an isometry.
They turn out to be quotients of a real Lie group by a maximal compact subgroup whose Lie algebra is the fixed subalgebra of the involution obtained by differentiating the geodesic symmetry.
This algebraic data is enough to provide a classification of the Riemannian symmetric spaces.