Glossary of Riemannian and metric geometry
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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:
- 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 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.
A
editAlexandrov 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]
B
editBarycenter, 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
C
editCartan–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
Such a point is unique if all distances are less than the convexity radius.
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 are called conjugate if there is a Jacobi field on which has a zero at p and q.
Convex function. A function f on a Riemannian manifold is a convex if for any geodesic the function is convex. A function f is called -convex if for any geodesic with natural parameter , the function is convex.
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.[3] Sometimes the additional requirement is made that the distance function to in these balls is convex.[4]
D
editDiameter 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
E
editExponential map: Exponential map (Lie theory), Exponential map (Riemannian geometry)
F
editFirst fundamental form for an embedding or immersion is the pullback of the metric tensor.
G
editGeodesic 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 where is a geodesic.
Geodesic metric space is a metric space where any two points are the endpoints of a minimizing geodesic.
H
editHadamard space is a complete simply connected space with nonpositive curvature.
Horosphere a level set of Busemann function.
I
editInjectivity 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.[5] 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.[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 on N. An orbit space of N by a discrete subgroup of which acts freely on N is called an infranilmanifold. An infranilmanifold is finitely covered by a nilmanifold.[7]
Isometry is a map which preserves distances.
J
editJacobi 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 with , then the Jacobi field is described by
K
editL
editLength 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).[8]
Logarithmic map, or logarithm, is a right inverse of Exponential map.[9][10]
M
editMinimal surface is a submanifold with (vector of) mean curvature zero.
N
editNatural parametrization is the parametrization by length.[11]
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 .[12] This is distinct from topological nets which generalize limits.
Nilmanifold: An element of the minimal set of manifolds which includes a point, and has the following property: any oriented -bundle over a nilmanifold is a nilmanifold. 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 , 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.
P
editPolyhedral 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.[13]
Q
editQuasigeodesic has two meanings; here we give the most common. A map (where is a subinterval) is called a quasigeodesic if there are constants and such that for every
Note that a quasigeodesic is not necessarily a continuous curve.
Quasi-isometry. A map is called a quasi-isometry if there are constants and such that
and every point in Y has distance at most C from some point of f(X). Note that a quasi-isometry is not assumed to be continuous. For example, any map between compact metric spaces is a quasi isometry. If there exists a quasi-isometry from X to Y, then X and Y are said to be quasi-isometric.
R
editRadius of metric space is the infimum of radii of metric balls which contain the space completely.[14]
Ray is a one side infinite geodesic which is minimizing on each interval.[15]
Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same time.
S
editSecond 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.
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
(there is no standard agreement whether to use + or − in the definition).
Short map is a distance non increasing map.
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[16] 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.
T
editTotally 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.[17]
Totally geodesic submanifold is a submanifold such that all geodesics in the submanifold are also geodesics of the surrounding manifold.[18]
U
editUniquely geodesic metric space is a metric space where any two points are the endpoints of a unique minimizing geodesic.
W
editWord metric on a group is a metric of the Cayley graph constructed using a set of generators.
References
edit- ^ Kobayashi, Shōshichi; Nomizu, Katsumi (1963). "Chapter VII Submanifolds, 8. Autoparallel submanifolds and totally geodesic submanifolds". Foundations of differential geometry. Interscience Publishers, New York, NY. pp. 53–62. ISBN 978-0-471-15732-8. Zbl 0175.48504.
- ^ Mancinelli, Claudio; Puppo, Enrico (2023-06-01). "Computing the Riemannian center of mass on meshes". Computer Aided Geometric Design. 103: 102203. doi:10.1016/j.cagd.2023.102203. ISSN 0167-8396.
- ^ Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (eds.), "Riemannian metrics", Riemannian Geometry, Berlin, Heidelberg: Springer, Remark after Proof of Corollary 2.89, p.87, doi:10.1007/978-3-642-18855-8_2, ISBN 978-3-642-18855-8, retrieved 2024-11-28
- ^ Petersen, Peter (2016), Petersen, Peter (ed.), "Sectional Curvature Comparison I", Riemannian Geometry, Cham: Springer International Publishing, Theorem 6.4.8, pp. 258-259, doi:10.1007/978-3-319-26654-1_6, ISBN 978-3-319-26654-1, retrieved 2024-11-29
- ^ Lee, Jeffrey M. (2009). "13. Riemannian and Semi-Riemannian Geometry, Definition 13.141". Manifolds and differential geometry. Providence, RI: American Mathematical Society (AMS). p. 615. ISBN 978-0-8218-4815-9. Zbl 1190.58001.
- ^ Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (eds.), "Curvature", Riemannian Geometry, Berlin, Heidelberg: Springer, Scholium 3.78, doi:10.1007/978-3-642-18855-8_3, ISBN 978-3-642-18855-8, retrieved 2024-11-28
- ^ Hirsch, Morris W. (1970). "Expanding maps and transformation groups". Global Analysis, Proc. Sympos. Pure Math. (14): 125–131. doi:10.1090/pspum/014/0298701. Zbl 0223.58009.
- ^ Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 7, §7.2, pp. 249-250. ISBN 0-8218-2129-6. Zbl 0981.51016.
- ^ Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 9, §9.1, pp. 321-322. ISBN 0-8218-2129-6. Zbl 0981.51016.
- ^ Lang, Serge (1999). "Fundamentals of Differential Geometry". Graduate Texts in Mathematics. Chapter XII An example of seminegative curvature, p. 323. doi:10.1007/978-1-4612-0541-8. ISSN 0072-5285.
- ^ Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 2, §2.5.1, Definition 2.5.7. ISBN 0-8218-2129-6. Zbl 0981.51016.
- ^ Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 1, §1.6, Definition 1.6.1, p. 13. ISBN 0-8218-2129-6. Zbl 0981.51016.
- ^ Bridson, Martin R.; Haefliger, André (1999), Bridson, Martin R.; Haefliger, André (eds.), "Basic Concepts", Metric Spaces of Non-Positive Curvature, Berlin, Heidelberg: Springer, Chapter I.1, § Metric spaces, Definitions 1.1, p. 2, doi:10.1007/978-3-662-12494-9_1, ISBN 978-3-662-12494-9, retrieved 2024-11-29
- ^ Burago, Dmitri; Burago, Yurii; Ivanov, Sergei (2001). A course in metric geometry. Providence, RI: American Mathematical Society (AMS). Chapter 10, §10.4, Exercise 10.4.5, p. 366. ISBN 0-8218-2129-6. Zbl 0981.51016.
- ^ Petersen, Peter (2016). "Riemannian Geometry". Graduate Texts in Mathematics. Chapter 7, §7.3.1 Rays and Lines, p. 298. doi:10.1007/978-3-319-26654-1. ISSN 0072-5285.
- ^ Berestovskii, V. N. (1987-07-01). "Submetries of space-forms of negative curvature". Siberian Mathematical Journal. 28 (4): 552–562. doi:10.1007/BF00973842. ISSN 1573-9260.
- ^ Petersen, Peter (2016). "Riemannian Geometry". Graduate Texts in Mathematics. Chapter 12, §12.4 The Soul Theorem, p. 463. doi:10.1007/978-3-319-26654-1. ISSN 0072-5285.
- ^ Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004). "Riemannian Geometry". Universitext. Chapter 2, §2.C.1, Definition 2.80 bis, p.82. doi:10.1007/978-3-642-18855-8. ISSN 0172-5939.