Science Fair Project Encyclopedia
Glossary of Riemannian and metric geometry
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. These either contain specialised vocabulary or provide more detailed expositions of the definitions given below.
- Glossary of general topology
- Glossary of differential geometry and topology
- List of differential geometry topics
Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or | xy | X denotes the distance between points x and y in X. Italic word denotes a self-reference to this glossary.
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.
Baricenter, see center of mass.
bi-Lipschitz map. A map is called bi-Lipschitz if there are positive constants c and C such that for any x and y in X
Busemann function given a ray, γ : [0, ∞)→X, the Busemann function is defined by
Center of mass. A point q∈M is called the center of mass of the points p1,p2,..,pk if it is a point of global minimum of the function
f(x) = ∑ | pix | 2 i
Such a point is unique if all distances | pipj | are less than radius of convexity.
Conformally flat a M is conformally flat if it is locally conformally equivalent to a Euclidean space, for example standard sphere is conformally flat.
Conjugate point two points p and q on a geodesic γ are called conjugate if there is a Jacobi field on γ which has a zero at p and q.
Convex function. A function f on a Riemannian manifold is a convex if for any geodesic γ the function is convex. A function f is called λ-convex if for any geodesic γ with natural parameter t, the function is convex.
Convex A subset K of metric space M is called convex if for any two points in K there is a shortest path connecting them which lies entirely in K, see also totally convex.
Diameter of a metric space is the supremum of distances between pairs of points.
Dilation of a map between metric spaces is the infimum of numbers L such that the given map is L-Lipschitz.
Hadamard space is a complete simply connected space with nonpositive curvature.
Horosphere a level set of Busemann function.
Injectivity radius at a point p of a Riemannian manifold is the largest radius for which the exponential map at p is a diffeomorphism. Injectivity radius of a Riemannian manifold is the infimum of Injectivity radii at all points.
For complete manifolds, if the injectivity radius at p is a finite, say 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 and on the distance r from p. For closed Riemannian manifold the injectivity radius is either half of minimal length of closed geodesic or minimal distance between conjugate points on a geodesic.
Infranil manifold 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 semidirect product NF on N. A compact factor of N by subgroup of NF acting freely on N is called infranil manifold. Infranil manifolds are factors of nill manifolds by finite group (but wiseversa it is not longer true).
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 γτ with γ0 = γ, then the Jacobi field
Length metric the same as intrinsic metric.
Levi-Civita connection is a natural way to differentiate vector field on Riemannian manifolds.
Lipschitz convergence the convergence defined by Lipsitz metric.
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).
Logarithmic map is a right inverse of Exponential map
Minimal surface is a submanifold with (vector of) mean curvature zero.
Natural parametrization is the parametrization by length
Net. A sub set S of a metric space X is called ε-net if for any point in X there is a point in S on the distance . This is distinct from topological nets which generalise limits.
Nil manifolds: the minimal set of manifolds which includes a point, and has the following property: any oriented S1-bundle over a nil manifold is a nil manifold. It also can be defined as a factor of a connected nilpotent Lie group by a lattice.
Nonexpanding map same as short map
Quasigeodesic. has two meanings here is the most common meaning: A map f:R is called quasigeodesic if there is a constant C such that
Note that quasigeodesic is not a continuous curve in general.
Quasi-isometry. A map is called a quasi-isometry if there is a constant C such that f(X) is a C-net in Y and
Note that quasi isometry is not assumed to be continuous, for example any map between compact metric spaces is a quasi isometry.
Radius of metric space is the infimum of radii of metric balls which contain the space completely.
Radius of convexity at a point p of a Riemannian manifold is the largest radius of a ball which is a convex subset.
Ray is a one side infinite geodesic which is minimizing on each interval
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 shape operator of a hypersurface,
It can be also generalized to arbitrary codimension, then it is a quadratic form with values in the normal space.
Shape operator for a hypersurface M is a linear operator from . 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).
Short map is a distance non increasing map.
Submetry a short map f between metric spaces called submetry if for any point x and radius r we have that image of metric r-ball is an r-ball, i.e.
- f(Br(x)) = Br(f(x))
Systole. k-systole of M, systk(M), is the minimal volume of k-cycle nonhomologous to zero.
Totally convex. A subset K of metric space M is called totally convex if for any two points in K any shortest path connecting them lies entirely in K, see also convex.
Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.
Word metric on a group is a metric of the Cayley graph constructed using a set of generators.
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