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In mathematics, **constant curvature** is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Differential geometry** is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

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 plane σ_{p} in the tangent space at a point *p* of the manifold. It is the Gaussian curvature of the surface which has the plane σ_{p} as a tangent plane at *p*, obtained from geodesics which start at *p* in the directions of σ_{p}. The sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

The Riemannian manifolds of constant curvature can be classified into the following three cases:

In differential geometry, a (**smooth**) **Riemannian manifold** or (**smooth**) **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with an inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p* that varies smoothly from point to point in the sense that if *X* and *Y* are differentiable vector fields on *M*, then *p* ↦ *g*_{p}(*X*|_{p}, *Y*|_{p}) is a smooth function. The family *g*_{p} of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

- elliptic geometry – constant positive sectional curvature
- Euclidean geometry – constant vanishing sectional curvature
- hyperbolic geometry – constant negative sectional curvature.

**Elliptic geometry** is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as *single elliptic geometry* whereas spherical geometry is sometimes referred to as *double elliptic geometry*.

**Euclidean geometry** is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The *Elements* begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the *Elements* states results of what are now called algebra and number theory, explained in geometrical language.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

- Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel .
- Every space of constant curvature is locally maximally symmetric, i.e. it has number of local isometries, where n is its dimension.
- Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has (global) isometries, has constant curvature.
- (Killing–Hopf theorem) The universal cover of a manifold of constant sectional curvature is one of the model spaces:
- sphere (sectional curvature positive)
- plane (sectional curvature zero)
- hyperbolic manifold (sectional curvature negative)

A

**sphere**is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.In mathematics, a Riemannian manifold is said to be

**flat**if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.In mathematics, a

**hyperbolic space**is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature. - A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details).
- Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same signature and their sectional curvatures are equal.

In the mathematical field of differential geometry, the **Riemann curvature tensor** or **Riemann–Christoffel tensor** is the most common method used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold, that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In geometry, **parallel** lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

In geometry, the **Killing–Hopf theorem** states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space forms. The Killing–Hopf theorem was proved by Killing (1891) and Hopf (1926).

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In geometry, a **pseudosphere** is a surface with constant negative Gaussian curvature. Hilbert's theorem says that no pseudosphere can be immersed into three-dimensional space.

In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, 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 and volume. From those, some other global quantities can be derived by integrating local contributions.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it 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. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In differential geometry, the **Gaussian curvature** or **Gauss curvature** *Κ* of a surface at a point is the product of the principal curvatures, *κ*_{1} and *κ*_{2}, at the given point:

In differential geometry, the **Ricci flow** is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.

In mathematics and physics, *n*-dimensional **anti-de Sitter space** (AdS_{n}) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.

In mathematics and especially differential geometry, a **Kähler manifold** is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

In differential geometry and mathematical physics, an **Einstein manifold** is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations, although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

In mathematics, a **space form** is a complete Riemannian manifold *M* of constant sectional curvature *K*. The three obvious examples are Euclidean *n*-space, the *n*-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

In mathematics, a **hyperbolic manifold** is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman.

In mathematics, an **isotropic manifold** is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold
is isotropic if for any point
and unit vectors
, there is an isometry
of
with
and
. Every complete isotropic manifold is homogeneous, i.e. for any
there is an isometry
of
with
This can be seen by considering a geodesic
from
to
and taking the isometry which fixes
and maps
to

In differential geometry, a subfield of mathematics, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds. Roughly, it states that within a fixed radius, usually called the **Margulis constant**, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

In differential geometry, **Hilbert's theorem** (1901) states that there exists no complete regular surface
of constant negative gaussian curvature
immersed in
. This theorem answers the question for the negative case of which surfaces in
can be obtained by isometrically immersing complete manifolds with constant curvature.

In mathematics, spaces of **non-positive curvature** occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces.

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 perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.