Riemannian geometry
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Riemannian geometry is the branch of
Riemannian geometry originated with the vision of
Introduction
Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose
Every smooth manifold admits a
There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature.[2][3]
The following articles provide some useful introductory material:
- Metric tensor
- Riemannian manifold
- Levi-Civita connection
- Curvature
- Riemann curvature tensor
- List of differential geometry topics
- Glossary of Riemannian and metric geometry
Classical theorems
What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below).
The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.
General theorems
- generalized Gauss-Bonnet theorem.
- Nash embedding theorems. They state that every Riemannian manifold can be isometrically embedded in a Euclidean spaceRn.
Geometry in large
In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.
Pinched sectional curvature
- Sphere theorem. If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere.
- Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D and volume ≥ V.
- nil manifold.
Sectional curvature bounded below
- Cheeger–Gromoll's G. Perelmanin 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: M is diffeomorphic to Rn if it has positive curvature at only one point.
- Gromov's Betti number theorem. There is a constant C = C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C.
- Grove–Petersen's finiteness theorem. Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.
Sectional curvature bounded above
- The diffeomorphic to the Euclidean space Rn with n = dim M via the exponential mapat any point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
- The ergodic.
- If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT(k) space. Consequently, its fundamental group Γ = π1(M) is Gromov hyperbolic. This has many implications for the structure of the fundamental group:
- it is finitely presented;
- the word problem for Γ has a positive solution;
- the group Γ has finite virtual cohomological dimension;
- it contains only finitely many conjugacy classes of elements of finite order;
- the virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.
- it is
Ricci curvature bounded below
- Myers theorem. If a complete Riemannian manifold has positive Ricci curvature then its fundamental groupis finite.
- Bochner's formula. If a compact Riemannian n-manifold has non-negative Ricci curvature, then its first Betti number is at most n, with equality if and only if the Riemannian manifold is a flat torus.
- Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold that has nonnegative Ricci curvature.
- Bishop–Gromov inequality. The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.
- Gromov-Hausdorff metric.
Negative Ricci curvature
- The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.
- Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.[4] (This is not true for surfaces.)
Positive scalar curvature
- The n-dimensional torus does not admit a metric with positive scalar curvature.
- If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n(n-1).
See also
- Shape of the universe
- Introduction to the mathematics of general relativity
- Normal coordinates
- Systolic geometry
- Riemann–Cartan geometryin Einstein–Cartan theory (motivation)
- Riemann's minimal surface
- Reilly formula
Notes
- ^ maths.tcd.ie
- ^
Kleinert, Hagen (1989). "Gauge Fields in Condensed Matter Vol II": 743–1440.
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Bibcode:2008mfcm.book.....K.
- ^ Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.
References
- Books
- ISBN 0-8218-2052-4. (Provides a historical review and survey, including hundreds of references.)
- Cheeger, Jeff; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, Providence, RI: AMS Chelsea Publishing; Revised reprint of the 1975 original.
- Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Riemannian geometry, Universitext (3rd ed.), Berlin: Springer-Verlag.
- Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag, ISBN 3-540-42627-2.
- Petersen, Peter (2006), Riemannian Geometry, Berlin: Springer-Verlag, ISBN 0-387-98212-4
- From Riemann to Differential Geometry and Relativity (Lizhen Ji, Athanase Papadopoulos, and Sumio Yamada, Eds.) Springer, 2017, XXXIV, 647 p. ISBN 978-3-319-60039-0
- Papers
- S2CID 15463483
External links
- Riemannian geometry by V. A. Toponogov at the Encyclopedia of Mathematics
- Weisstein, Eric W. "Riemannian Geometry". MathWorld.