All Topics  
Riemannian geometry

 

 
   Email Print
   Bookmark   Link
 

 

Riemannian geometry
 
 
  Topics Riemannian geometry Topic Home

Elliptic geometry
Elliptic geometry

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a Point p outside L, there exists no line Parallel to L passing through p....
 is also sometimes called "Riemannian geometry".


Riemannian geometry is the branch of differential geometry that studies Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
s, smooth manifolds
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 with a Riemannian metric, i.e. with an inner product on the tangent space
Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
 at each point which varies smoothly
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
 from point to point. This gives in particular local notions of angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
, length of curves
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
, surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
, and volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. From those some other global quantities can be derived by integrating
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann
Bernhard Riemann

Georg Friedrich Bernhard Riemann was a Germany mathematics who made important contributions to mathematical analysis and differential geometry, some of them paving the way for the later development of general relativity....
 expressed in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (German: On the hypotheses on which geometry is based).






Discussion
Ask a question about 'Riemannian geometry'
Start a new discussion about 'Riemannian geometry'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Elliptic geometry
Elliptic geometry

Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a Point p outside L, there exists no line Parallel to L passing through p....
 is also sometimes called "Riemannian geometry".


Riemannian geometry is the branch of differential geometry that studies Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
s, smooth manifolds
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 with a Riemannian metric, i.e. with an inner product on the tangent space
Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
 at each point which varies smoothly
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
 from point to point. This gives in particular local notions of angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
, length of curves
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
, surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
, and volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
. From those some other global quantities can be derived by integrating
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
 local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann
Bernhard Riemann

Georg Friedrich Bernhard Riemann was a Germany mathematics who made important contributions to mathematical analysis and differential geometry, some of them paving the way for the later development of general relativity....
 expressed in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (German: On the hypotheses on which geometry is based). It is a very broad and abstract generalization of the differential geometry of surfaces
Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth manifold surfaces with various additional structures, most often, a Riemannian metric....
 in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
s on them, with techniques that can be applied to the study of differentiable manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
s of higher dimensions. It inspired Einstein
Albert Einstein

Albert Einstein was a Germany-born theoretical physics. He is best known for his theory of relativity and specifically mass?energy equivalence, expressed by the equation E = mc2....
's general relativity theory, made profound impact on group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and representation theory
Representation theory

Representation theory is a branch of mathematics that studies abstract algebra algebraic structures by representing their element as linear transformations of vector spaces....
, as well as analysis, and spurred the development of algebraic
Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant that classification theorem topological spaces up to homeomorphism....
 and differential topology
Differential topology

In mathematics, differential topology is the field dealing with differentiable function s on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds....
.

Introduction

Georg Friedrich Bernhard Riemann
Riemannian geometry was first put forward in generality by Bernhard Riemann
Bernhard Riemann

Georg Friedrich Bernhard Riemann was a Germany mathematics who made important contributions to mathematical analysis and differential geometry, some of them paving the way for the later development of general relativity....
 in the nineteenth century. It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean geometry
Non-Euclidean geometry

In mathematics, non-Euclidean geometry describes hyperbolic geometry and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of Parallel lines....
, spherical geometry
Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
 and hyperbolic geometry
Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
, as well as Euclidean geometry
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
 itself.

Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology
Differential topology

In mathematics, differential topology is the field dealing with differentiable function s on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds....
. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifold
Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann....
s, which (in four dimensions) are the main objects of the theory of general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
. Other generalizations of Riemannian geometry include Finsler geometry
Finsler manifold

In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M with a Banach norm defined over each tangent space, smooth function depending on position, and assumed to satisfy the following condition:...
 and spray spaces.

There is no easy introduction to Riemannian geometry. It is generally recommended that one should work in the subject for quite a while to build some geometric intuition, usually by doing enormous amounts of calculations. The following articles might serve as a rough introduction:

  1. Metric tensor
    Metric tensor

    In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
  2. Riemannian manifold
    Riemannian manifold

    In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
  3. Levi-Civita connection
    Levi-Civita connection

    In Riemannian geometry, the Levi-Civita connection is the Torsion -free Riemannian connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric....
  4. Curvature
    Curvature

    In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
  5. Curvature tensor
    Curvature tensor

    The term curvature tensor is ambiguous in its generality. It could refer to:* the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;...
    .


The following articles might also be useful:

  1. List of differential geometry topics
    List of differential geometry topics

    This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics....
  2. Glossary of Riemannian and metric geometry
    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....


Classical theorems in Riemannian geometry


What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance, beauty, and simplicity of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger
Jeff Cheeger

Jeff Cheeger , is a mathematician. Cheeger is professor at the Courant Institute at New York University in New York City. His main interests are differential geometry and its applications to topology and Mathematical analysis....
 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


  1. Gauss–Bonnet theorem
    Gauss–Bonnet theorem

    The Gauss?Bonnet theorem or Gauss?Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology ....
     The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to where denotes the Euler characteristic
    Euler characteristic

    In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
     of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem
    Generalized Gauss-Bonnet theorem

    In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature....
    .
  2. Nash embedding theorem
    Nash embedding theorem

    The Nash embedding theorems , named after John Forbes Nash, state that every n-dimensional Riemannian manifold can be isometrically embedding in a Euclidean space Rm....
    s     also called fundamental theorems of Riemannian geometry
    Fundamental theorem of Riemannian geometry

    In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique torsion metric affine connection, called the Levi-Civita connection of the given metric....
    . They state that every Riemannian manifold
    Riemannian manifold

    In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
     can be isometrically embedded
    Embedding

    In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
     in a Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
     Rn.


Local to global theorems


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

  1. Brendle and Schoen's differential sphere theorem. If M is a compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a spherical space form. This is sharp: complex projective space
    Complex projective space

    In mathematics, complex projective space, P, Pn or CPn, in fact preferablyis the projective space of line in Cn+1....
     has curvature non-strictly pinched between 1/4 and 1. If strict pinching is replaced by weak pinching (i.e. if the sectional curvature of M lies in the closed interval ), then M is diffeomorphic to a spherical space form or isometric to a locally symmetric space. For more information see the article on the Sphere theorem
    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....
    .
  2. 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 , diameter and volume .
  3. 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 , diameter and volume .
  4. Gromov's almost flat manifolds
    Almost flat manifold

    In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e....
    .     There is an such that if an n-dimensional Riemannian manifold has a metric with sectional curvature and diameter then its finite cover is diffeomorphic to a nil manifold.


Positive sectional curvature

  1. Cheeger-Gromoll's Soul theorem
    Soul theorem

    In mathematics, the soul theorem is the following theorem of Riemannian geometry:The submanifold S is called a soul of . The soul is not uniquely determined, but any two souls are isometric....
    .     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 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 Rn. G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: M is diffeomorphic Rn if it has positive curvature at only one point.
  2. 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
    Betti number

    In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces....
    s is at most C.


Positive Ricci curvature
Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space....
 

  1. Myers theorem
    Myers theorem

    The Myers theorem, also known as the Bonnet-Myers theorem, is a classical theorem in Riemannian geometry. The strong form was proven by ....
    .     If a compact Riemannian manifold has positive Ricci curvature then its fundamental group
    Fundamental group

    In mathematics, more specifically algebraic topology, the fundamental group or Poincar? group is a group associated to any given pointed space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other....
     is finite.
  2. Cheeger-Gromoll Splitting theorem
    Splitting theorem

    The splitting theorem is a classical theorem in Riemannian geometry. It states that if a complete Riemannian manifold M with Ricci curvature...
    .     If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic which 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 which has nonnegative Ricci curvature.
  3. Bishop's 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.
  4. Gromov's compactness theorem
    Gromov's compactness theorem (geometry)

    In Riemannian geometry, Gromov's compactness theorem states thatthe set of Riemannian manifolds of a given dimension, with Ricci curvature = c and diameter = D is relatively compact in the Gromov-Hausdorff metric....
    .     The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact
    Metric space

    In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
     in the Gromov-Hausdorff metric
    Gromov-Hausdorff convergence

    Gromov?Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff distance....
    .


Positive scalar curvature

  1. The n-dimensional torus does not admit a metric with positive scalar curvature.
  2. If the injectivity radius
    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....
     of a compact n-dimensional Riemannian manifold is then the average scalar curvature is at most n(n-1).


Non-positive sectional curvature

  1. The Cartan–Hadamard theorem
    Cartan–Hadamard theorem

    The Cartan?Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature....
     states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space
    Euclidean space

    Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
     R^n with n = dim M via the exponential map
    Exponential map

    In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
     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.


Negative sectional curvature

  1. The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.
  2. 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
    CAT(k) space

    In mathematics, a CAT space is a specific type of metric space. Intuitively, triangles in a CAT space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k....
    . Consequently, its fundamental group
    Fundamental group

    In mathematics, more specifically algebraic topology, the fundamental group or Poincar? group is a group associated to any given pointed space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other....
     G = p1(M) is Gromov hyperbolic
    Hyperbolic group

    In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry....
    . This has many implications for the structure of the fundamental group:
  • it is finitely presented;
  • the word problem
    Word problem for groups

    In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a presentation of a group group G is the algorithmic problem of deciding whether two words represent the same element....
     for Γ has a positive solution;
  • the group Γ has finite virtual cohomological dimension
    Cohomological dimension

    In abstract algebra, cohomological dimension is an invariant which measures the homological complexity of representations of a group . It has important applications in geometric group theory, topology, and algebraic number theory....
    ;
  • it contains only finitely many conjugacy class
    Conjugacy class

    In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
    es of elements of finite order;
  • the abelian
    Abelian group

    An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
     subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.


Negative Ricci curvature

  1. The isometry group
    Isometry

    In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
     of a compact Riemannian manifold with negative Ricci curvature is discrete
    Discrete group

    In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one....
    .
  2. Any smooth manifold of dimension admits a Riemannian metric with negative Ricci curvature. (This is not true for surfaces.)


See also

  • Shape of the universe
    Shape of the Universe

    The shape of the Universe is an informal name for a subject of investigation within physical cosmology which describes the geometry of the universe including both #Local geometry and #Global geometry....
  • Basic introduction to the mathematics of curved spacetime
    Basic introduction to the mathematics of curved spacetime

    An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. In order to understand special relativity one also needs an understanding of tensor calculus....
  • Normal coordinates
    Normal coordinates

    In Riemannian geometry, the normal coordinates at p consist of a chart such that locally the symmetric part of the Christoffel symbols vanish, i.e....
  • Systolic geometry


External links