In differential geometry, **Riemannian geometry** is the study of smooth manifolds with Riemannian metrics, i.e. a choice of positive-definite quadratic form on a manifold's tangent spaces which varies smoothly from point to point. This gives in particular local ideas of angle, length of curves, and volume. From those some other global quantities can be derived by integrating local contributions. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...
On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
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. ...
An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
## Introduction
Riemannian geometry was first put forward in generality by Bernhard Riemann 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, spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself. Bernhard Riemann. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ...
Spherical geometry is the geometry of the two-dimensional surface of a sphere. ...
Lines through a given point P and hyperparallel to line l. ...
Euclid 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. 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 mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...
General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...
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: - Metric tensor
- Riemannian manifold
- Levi-Civita connection
- Curvature
- Curvature tensor.
The following articles might also be useful: In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...
Curvature refers to a number of loosely related concepts in different areas of geometry. ...
In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...
- List of differential geometry topics
- Glossary of Riemannian and metric geometry
This is a list of differential geometry topics, by Wikipedia page. ...
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...
## 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. 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 of *M*. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem. **Nash embedding theorems** also called fundamental theorems of Riemannian geometry. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space **R**^{n}. The Gaussâ€“Bonnet theorem or Gaussâ€“Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...
It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed Riemannian manifold as an integral of a certain polynomial derived from its curvature. ...
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. ...
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. ...
In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
### 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 **1/4-pinched sphere theorem.** If *M* is a complete *n*-dimensional Riemannian manifold with sectional curvature strictly pinched between 1 and 4 then *M* is homeomorphic to *n*-sphere. **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 and diameter . **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 and diameter then its finite cover is diffeomorphic to a nil 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. ...
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...
#### Positive curvature ##### Positive sectional curvature **Soul theorem.** If *M* is a non-compact complete positively curved *n*-dimensional Riemannian manifold then it is diffeomorphic to **R**^{n}. **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 mathematics, the soul theorem is the following theorem of Riemannian geometry: If (M,g) is a complete non-compact Riemannian manifold with sectional curvature K â‰¥ 0, then (M,g) has a compact totally convex, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S...
In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
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. ...
##### Positive Ricci curvature **Myers theorem.** If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite. **Splitting theorem.** 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 **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. **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. The Myers theorem, also known as the Bonnet-Myers theorem,is a classical theorem in Riemannian geometry. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
The splitting theorem is a classical theorem in Riemannian geometry. ...
In mathematics, the Bishop-Gromov inequality is a classical theorem in Riemannian geometry. ...
In Riemannian geometry, Gromovs compactness theorem states that the set of Riemannian manifolds with Ricci curvature â‰¥ c and diameter â‰¤ D is pre-compact in the Gromov-Hausdorff metric. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. ...
##### 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 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 — it doesnt cover the terminology of differential topology. ...
#### Negative curvature ##### Negative sectional curvature - The geodesic flow of any compact manifold with negative sectional curvature is ergodic.
- Any two points of a complete simply connected Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
- If
*M* is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of the fundamental group of *M* is isomorphic to **Z**. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
##### Negative Ricci curvature - Any compact Riemannian manifold with negative Ricci curvature has a discrete isometry group.
- Any smooth manifold of dimension admits a Riemannian metric with negative Ricci curvature.
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
## See also The shape of the Universe is an informal name for a subject of investigation within physical cosmology. ...
This article is on the minimal body of mathematics necessary to understand general relativity. ...
## External links ## References - Marcel Berger,
*Riemannian Geometry During the Second Half of the Twentieth Century*, (2000) University Lecture Series vol. 17, American Mathematical Society, Rhode Island, ISBN 0-8218-2052-4. *(Provides a historical review and survey, including hundreds of references.)* - Jurgen Jost,
*Riemannian Geometry and Geometric Analysis*, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 *(Provides a formal introduction, written at the grad-student level.)* - Peter Petersen,
*Riemannian Geometry*, (1998) Springer-Verlag, Berlin ISBN 0-387-98212-4. *(Provides an introduction, presented at a graduate level.)* |