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Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. Curved spaces can generally be described by
Riemannian geometry 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 ...
though some simple cases can be described in other ways. Curved spaces play an essential role in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, where
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
is often visualized as curved space. The
Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
is a curved metric which forms the current foundation for the description of the
expansion of space The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
and
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 ...
.


Simple two-dimensional example

A very familiar example of a curved space is the surface of a sphere. While to our familiar outlook the sphere ''looks'' three-dimensional, if an object is constrained to lie on the surface, it only has two dimensions that it can move in. The surface of a sphere can be completely described by two dimensions since no matter how rough the surface may appear to be, it is still only a surface, which is the two-dimensional outside border of a volume. Even the surface of the Earth, which is fractal in complexity, is still only a two-dimensional boundary along the outside of a volume.


Embedding

One of the defining characteristics of a curved space is its departure from the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
. In a curved space :dx^2 + dy^2 \neq dl^2. The Pythagorean relationship can often be restored by describing the space with an extra dimension. Suppose we have a non-euclidean three-dimensional space with coordinates \left(x',y',z'\right). Because it is not flat :dx'^2 + dy'^2 + dz'^2 \ne dl'^2 \,. But if we now describe the three-dimensional space with ''four'' dimensions (x,y,z,w) we can ''choose'' coordinates such that :dx^2 + dy^2 + dz^2 + dw^2 = dl^2 \,. Note that the coordinate x is not the same as the coordinate x'. For the choice of the 4D coordinates to be valid descriptors of the original 3D space it must have the same number of
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
. Since four coordinates have four degrees of freedom it must have a constraint placed on it. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. That is :x^2 + y^2 + z^2 +w^2 = \textrm \,. The constant can be positive or negative. For convenience we can choose the constant to be :\kappa^R^2 where R^2 \, now is positive and \kappa \equiv \plusmn 1. We can now use this constraint to eliminate the artificial fourth coordinate w. The differential of the constraining equation is :xdx + ydy + zdz + wdw = 0 \, leading to dw = -w^(xdx + ydy +zdz) \,. Plugging dw into the original equation gives :dl^2 = dx^2 + dy^2 + dz^2 + \frac. This form is usually not particularly appealing and so a coordinate transform is often applied: x = r\sin\theta\cos\phi, y = r\sin\theta\sin\phi, z = r\cos\theta. With this coordinate transformation :dl^2 = \frac + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.


Without embedding

The geometry of a n-dimensional space can also be described with
Riemannian geometry 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 ...
. An
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
and
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
space can be described by the metric: :dl^2 = e^ + r^2d\theta^2 + r^2\sin^2\theta d\phi^2 \,. This reduces to
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 ...
when \lambda = 0. But a space can be said to be " flat" when the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tida ...
has all zero components. In three dimensions this condition is met when the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
(R_) is equal to the metric times the
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
(R, not to be confused with the R of the previous section). That is R_ = g_ R. Calculation of these components from the metric gives that :\lambda = -\frac\ln \left( 1 - k r^2 \right) where k \equiv \frac. This gives the metric: :dl^2 = \frac + r^2d\theta^2 + r^2\sin^2\theta d\phi^2. where k can be zero, positive, or negative and is not limited to ±1.


Open, flat, closed

An
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
and
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
space can be described by the metric: :dl^2 = \frac + r^2d\theta^2 + r^2\sin^2\theta d\phi^2. In the limit that the constant of curvature (R) becomes infinitely large, a flat,
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 ...
is returned. It is essentially the same as setting \kappa to zero. If \kappa is not zero the space is not Euclidean. When \kappa = +1 the space is said to be ''closed'' or
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
. When \kappa = -1 the space is said to be ''open'' or
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
. Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°. Triangles which lie on the surface of a closed space will have a sum of angles which is greater than 180°. The volume, however, is not (4/3)\pi r^3.


See also

* CAT(''k'') space * Non-positive curvature


Further reading

*


External links


Curved Spaces
simulator for multiconnected universes developed by Jeffrey Weeks {{DEFAULTSORT:Curved Space Riemannian geometry Physical cosmology Differential geometry General relativity