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Elliptic geometry is also sometimes called Riemannian geometry. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be e. with an inner product on the tangent space at each point which varies smoothly from point to point. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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 Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability This gives in particular local notions of angle, length of curves, surface area, and volume. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult Surface area is the measure of how much exposed Area an object has The volume of any solid 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 local contributions. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Riemannian geometry originated with the vision of Bernhard Riemann 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 in R3. In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. It inspired Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential

Contents

Introduction

Bernhard Riemann
Bernhard Riemann

Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar 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, spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Other generalizations of Riemannian geometry include Finsler geometry and spray spaces. In Mathematics, particularly Differential geometry, a Finsler manifold is a Differentiable manifold M with a Banach norm defined over

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
  2. Riemannian manifold
  3. Levi-Civita connection
  4. Curvature
  5. Curvature tensor. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry

The following articles might also be useful:

  1. List of differential geometry topics
  2. Glossary of Riemannian and metric geometry

Classical theorems in Riemannian geometry

What follows is an incomplete list of the most classical theorems in Riemannian geometry. This is a list of Differential geometry topics See also Glossary of differential and metric geometry and List of Lie group topics. This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology. The choice is made depending on its importance, beauty, and simplicity of formulation.

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 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. The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant This theorem has a generalization to any compact even-dimensional Riemannian manifold, see 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
  2. Nash embedding theorems also called fundamental theorems of Riemannian geometry. The Nash embedding theorems (or imbedding theorems) named after John Forbes Nash, state that every n-dimensional Riemannian manifold can be isometrically In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or Pseudo-Riemannian manifold) there is a They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

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. 1/4-pinched sphere theorem. If M is a complete, simply connected n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is homeomorphic to n-sphere. This is sharp: complex projective space has curvature non-strictly pinched between 1/4 and 1. In Mathematics, complex projective space, P ( C n +1 P n ( C) or CP n For more information see the article on the 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
  2. Cheeger's finiteness theorem. Given constants C and D there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature |K|\le C and diameter \le D.
  3. Gromov's almost flat manifolds. In Mathematics, a smooth Compact Manifold M is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon There is an εn > 0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K|\le \epsilon_n and diameter \le 1 then its finite cover is diffeomorphic to a nil manifold. In Mathematics, the definition of a nilmanifold has not been universally agreed upon

Positive sectional curvature

  1. Soul theorem. In Mathematics, the soul theorem is the following theorem of Riemannian geometry: If ( M, g) is a complete non If M is a non-compact complete positively curved n-dimensional Riemannian manifold then it is diffeomorphic to Rn. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable
  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 numbers is at most C. 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

Positive Ricci curvature

  1. Myers theorem. In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined The Myers theorem also known as the Bonnet-Myers theorem, is a classical theorem in Riemannian geometry. If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.
  2. Splitting theorem. The splitting theorem is a classical theorem in Riemannian geometry. 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. In Mathematics, the Bishop–Gromov inequality is a classical theorem in Riemannian geometry, named after Richard L 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. For Gromov's compactness theorem in symplectic topology see that article The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of Metric spaces which is a generalization of

Positive scalar curvature

  1. The n-dimensional torus does not admit a metric with positive scalar curvature.
  2. If the injectivity radius of a compact n-dimensional Riemannian manifold is \ge \pi then the average scalar curvature is at most n(n-1). This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology.

Non-positive sectional curvature

  1. The Cartan–Hadamard theorem states that the universal cover of a complete Riemannian manifold with nonpositive sectional curvature is diffeomorphic to a Euclidean space via the exponential map at any point. The Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive Sectional curvature In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine 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. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In Mathematics and Physics, the adjective ergodic is used to imply that a system satisfies the Ergodic hypothesis of Thermodynamics or that
  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. In Mathematics, a CAT( k) space is a specific type of Metric space. Consequently, its fundamental group Γ = π1(M) is Gromov hyperbolic. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated 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 abelian subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z. In Mathematics, one method of defining a group is by a presentation. In Mathematics, especially in the area of Abstract algebra known as Combinatorial group theory, the word problem for a recursively presented In Abstract algebra, cohomological dimension is an invariant which measures the homological complexity of representations of a group. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class In Abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

Negative Ricci curvature

  1. The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete. For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, a discrete group is a group G equipped with the Discrete topology.
  2. Any smooth manifold of dimension n \geq 3 admits a Riemannian metric with negative Ricci curvature[1]. (This is not true for surfaces. )

Notes

  1. ^ Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.

See also

References

External links

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