Riemannian geometry is the branch of differential geometry
that studies Riemannian manifold
s, smooth manifolds
with a ''Riemannian metric'', i.e. with an inner product
on the tangent space
at each point that varies smoothly
from point to point. This gives, in particular, local notions of angle
, length of curves
, surface area
. From those, some other global quantities can be derived by integrating
Riemannian geometry originated with the vision of Bernhard Riemann
expressed in his inaugural lectureUeber die Hypothesen, welche der Geometrie zu Grunde liegen
("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces
. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesic
s on them, with techniques that can be applied to the study of differentiable manifold
s of higher dimensions. It enabled the formulation of Einstein
's general theory of relativity
, made profound impact on group theory
and representation theory
, as well as analysis
, and spurred the development of algebraic
and differential topology
Riemannian geometry was first put forward in generality by Bernhard Riemann
in the 19th century. It deals with a broad range of geometries whose metric
properties vary from point to point, including the standard types of non-Euclidean geometry
Every smooth manifold admits a Riemannian metric
, which often helps to solve problems of differential topology
. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifold
s, which (in four dimensions) are the main objects of the theory of general relativity
. Other generalizations of Riemannian geometry include Finsler geometry
There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations
s produce torsions and curvature.
The following articles provide some useful introductory material:
*List of differential geometry topics
*Glossary of Riemannian and metric geometry
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.
The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(''M'') where χ(''M'') denotes the Euler characteristic
of ''M''. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem
#Nash embedding theorem
s. They state that every Riemannian manifold
can be isometrically embedded
in a Euclidean space
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
. 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''.
#Gromov's almost flat manifolds
. There is an ε''n''
> 0 such that if an ''n''-dimensional Riemannian manifold has a metric with sectional curvature |''K''| ≤ ε''n''
and diameter ≤ 1 then its finite cover is diffeomorphic to a nil manifold
Sectional curvature bounded below
#Cheeger–Gromoll's soul theorem
. If ''M'' is a non-compact complete non-negatively curved ''n''-dimensional Riemannian manifold, then ''M'' contains a compact, totally geodesic submanifold ''S'' such that ''M'' is diffeomorphic to the normal bundle of ''S'' (''S'' is called the soul of ''M''.) In particular, if ''M'' has strictly positive curvature everywhere, then it is diffeomorphic
. G. Perelman
in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: ''M'' is diffeomorphic to R''n''
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 number
s 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 Cartan–Hadamard theorem
states that a complete simply connected
Riemannian manifold ''M'' with nonpositive sectional curvature is diffeomorphic
to the Euclidean space
with ''n'' = dim ''M'' via the exponential map
at 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 geodesic flow
of any compact Riemannian manifold with negative sectional curvature is 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 = 1
(''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 class
es of elements of finite order
::* the abelian
subgroups of Γ are virtually cyclic
, so that it does not contain a subgroup isomorphic to Z×Z.
Ricci curvature bounded below
. If a compact Riemannian manifold has positive Ricci curvature then its fundamental group
. 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.
. 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.
. 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's compactness theorem
. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most ''D'' is pre-compact
in the 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.
[Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.]
(''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).
*Shape of the universe
*Basic introduction to the mathematics of curved spacetime
in Einstein–Cartan theory (motivation)
*Riemann's minimal surface
* . ''(Provides a historical review and survey, including hundreds of references.)''
* ; Revised reprint of the 1975 original.
* From Riemann to Differential Geometry and Relativity (Lizhen Ji, Athanase Papadopoulos, and Sumio Yamada, Eds.) Springer, 2017, XXXIV, 647 p.
by V. A. Toponogov at the Encyclopedia of Mathematics