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Riemannian geometry is the branch of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
that studies
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s, smooth manifolds with a ''Riemannian metric'', i.e. with an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at each point that varies smoothly from point to point. This gives, in particular, local notions of
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
, length of curves,
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
expressed in his inaugural lecture "''Ueber 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 In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s on them, with techniques that can be applied to the study of
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s of higher dimensions. It enabled the formulation of
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's
general theory of relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
, made profound impact on
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
and
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, as well as
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, and spurred the development of
algebraic Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
and
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
.


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
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
properties vary from point to point, including the standard types of
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
. Every smooth manifold admits a Riemannian metric, which often helps to solve problems of
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
. It also serves as an entry level for the more complicated structure of
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
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.
Dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to s ...
s and disclinations produce torsions and curvature. The following articles provide some useful introductory material: *
Metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
*
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
*
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
*
Curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
*
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
* List of differential geometry topics *
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; they either contain specialised vocabulary or prov ...


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 Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry an ...
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

# Gauss–Bonnet theorem 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of ''M''. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. #
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
s. They state that every
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
can be isometrically embedded in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
R''n''.


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 In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a p ...

#
Sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. ...
. 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 to R''n''. 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 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 Cartan–Hadamard theorem states that a complete
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
Riemannian manifold ''M'' with nonpositive sectional curvature is diffeomorphic to the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
R''n'' 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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of any compact Riemannian manifold with negative sectional curvature is
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
. #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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
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 In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomologic ...
; ::* it contains only finitely many conjugacy classes of elements of finite order; ::* the
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.


Ricci curvature bounded below

# Myers theorem. If a complete Riemannian manifold has positive Ricci curvature then its
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
is finite. #
Bochner's formula In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner. Formal statement If u \colon M \ri ...
. 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 In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...
. 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 In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
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 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 prov ...
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 The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes ge ...
* Basic introduction to the mathematics of curved spacetime *
Normal coordinates In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of ''p'' obtained by applying the exponential map to the tange ...
*
Systolic geometry In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and o ...
* Riemann–Cartan geometry in Einstein–Cartan theory (motivation) * Riemann's minimal surface * Reilly formula


Notes


References

;Books * . ''(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. ;Papers *


External links


Riemannian geometry
by V. A. Toponogov at the Encyclopedia of Mathematics * {{Authority control Bernhard Riemann