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 provide more detailed expositions of the definitions given below.
See also:
Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or
|xy|X
A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general mathematical usage.
Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2)
Arc-wise isometry the same as path isometry.
Autoparallel the same as totally geodesic.[1]
Barycenter, see center of mass.
bi-Lipschitz map. A map
f:X\toY
c|xy|X\le|f(x)f(y)|Y\leC|xy|X
Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by
B\gamma(p)=\limt\toinfty(|\gamma(t)-p|-t)
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.
Cartan extended Einstein's General relativity to Einstein–Cartan theory, using Riemannian-Cartan geometry instead of Riemannian geometry. This extension provides affine torsion, which allows for non-symmetric curvature tensors and the incorporation of spin–orbit coupling.
Center of mass. A point is called the center of mass[2] of the points if it is a point of global minimum of the function
f(x)=\sumi
| 2. | |
| |p | |
| ix| |
Such a point is unique if all distances
|pipj|
Completion
Conformal map is a map which preserves angles.
Conformally flat a manifold M is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.
Conjugate points two points p and q on a geodesic
\gamma
\gamma
Convex function. A function f on a Riemannian manifold is a convex if for any geodesic
\gamma
f\circ\gamma
λ
\gamma
t
f\circ\gamma(t)-λt2
Convex A subset K of a Riemannian manifold M is called convex if for any two points in K there is a unique shortest path connecting them which lies entirely in K, see also totally convex.
Convexity radius at a point of a Riemannian manifold is the supremum of radii of balls centered at that are (totally) convex. The convexity radius of the manifold is the infimum of the convexity radii at its points; for a compact manifold this is a positive number. Sometimes the additional requirement is made that the distance function to in these balls is convex.
Diameter of a metric space is the supremum of distances between pairs of points.
Developable surface is a surface isometric to the plane.
Dilation same as Lipschitz constant
Exponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)
First fundamental form for an embedding or immersion is the pullback of the metric tensor.
Geodesic is a curve which locally minimizes distance.
Geodesic flow is a flow on a tangent bundle TM of a manifold M, generated by a vector field whose trajectories are of the form
(\gamma(t),\gamma'(t))
\gamma
Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.
Hadamard space is a complete simply connected space with nonpositive curvature.
Horosphere a level set of Busemann function.
Injectivity radius The injectivity radius at a point p of a Riemannian manifold is the supremum of radii for which the exponential map at p is a diffeomorphism. The injectivity radius of a Riemannian manifold is the infimum of the injectivity radii at all points.[3] See also cut locus.
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. 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.
N\rtimesF
Isometry is a map which preserves distances.
Jacobi field A Jacobi field is a vector field on a geodesic γ which can be obtained on the following way: Take a smooth one parameter family of geodesics
\gamma\tau
\gamma0=\gamma
J(t)=\left.
| \partial\gamma\tau(t) | |
| \partial\tau |
\right|\tau=0.
Length metric the same as intrinsic metric.
Levi-Civita connection is a natural way to differentiate vector fields on Riemannian manifolds.
Lipschitz constant of a map is the infimum of numbers L such that the given map is L-Lipschitz.
Lipschitz convergence the convergence of metric spaces defined by Lipschitz distance.
Lipschitz distance between metric spaces is the infimum of numbers r such that there is a bijective bi-Lipschitz map between these spaces with constants exp(-r), exp(r).[5]
Logarithmic map, or logarithm, is a right inverse of Exponential map.[6] [7]
Metric ball
Minimal surface is a submanifold with (vector of) mean curvature zero.
Natural parametrization is the parametrization by length.[8]
Net. A subset S of a metric space X is called -net if for any point in X there is a point in S on the distance .[9] This is distinct from topological nets which generalize limits.
An element of the minimal set of manifolds which includes a point, and has the following property: any oriented
S1
associated to an embedding of a manifold M into an ambient Euclidean space , the normal bundle is a vector bundle whose fiber at each point p is the orthogonal complement (in ) of the tangent space .
Nonexpanding map same as short map.
Path isometry
Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space.
Principal curvature is the maximum and minimum normal curvatures at a point on a surface.
Principal direction is the direction of the principal curvatures.
Proper metric space is a metric space in which every closed ball is compact. Equivalently, if every closed bounded subset is compact. Every proper metric space is complete.
Quasigeodesic has two meanings; here we give the most common. A map
f:I\toY
I\subseteqR
K\ge1
C\ge0
x,y\inI
{1\overK}d(x,y)-C\led(f(x),f(y))\leKd(x,y)+C.
Quasi-isometry. A map
f:X\toY
K\ge1
C\ge0
{1\overK}d(x,y)-C\led(f(x),f(y))\leKd(x,y)+C.
Radius of metric space is the infimum of radii of metric balls which contain the space completely.[10]
Ray is a one side infinite geodesic which is minimizing on each interval.[11]
Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.
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,
II(v,w)=\langleS(v),w\rangle
Shape operator for a hypersurface M is a linear operator on tangent spaces, Sp: TpM→TpM. If n is a unit normal field to M and v is a tangent vector then
S(v)=\pm\nablavn
Short map is a distance non increasing map.
Smooth manifold
Sol manifold is a factor of a connected solvable Lie group by a lattice.
Submetry A short map f between metric spaces is called a submetry[12] 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
Systole. The k-systole of M, , is the minimal volume of k-cycle nonhomologous to zero.
Totally convex A subset K of a Riemannian manifold M is called totally convex if for any two points in K any geodesic connecting them lies entirely in K, see also convex.[13]
Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.[14]
Uniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic.
Word metric on a group is a metric of the Cayley graph constructed using a set of generators.