View source for Riemannian geometry

The theory of Riemannian spaces. A Riemannian space is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821501.png" />-dimensional connected [[Differentiable manifold|differentiable manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821502.png" /> on which a differentiable tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821503.png" /> of rank 2 is given which is covariant, symmetric and positive definite. The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821504.png" /> is called a metric tensor. Riemannian geometry is a multi-dimensional generalization of the intrinsic geometry (cf. [[Interior geometry|Interior geometry]]) of two-dimensional surfaces in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821505.png" />. The metric of a Riemannian space coincides with the Euclidean metric of the domain under consideration up to the first order of smallness. The difference between these metrics is (locally) estimated by the Riemannian curvature — a multi-dimensional generalization of the concept of the [[Gaussian curvature|Gaussian curvature]] of a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821506.png" />.

At the foundation of Riemannian geometry there are three ideas. The first of these is the realization of the fact that a non-Euclidean geometry exists — the geometry of N.I. Lobachevskii. The second is the concept of the interior geometry of surfaces created by C.F. Gauss. The third is the concept of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821507.png" />-dimensional space, developed in the first half of the 19th century by B. Riemann, who unified and generalized these ideas in his lecture  "On the hypotheses underlying the foundations of geometry"  [[#References|[1]]]. The concepts of Riemannian geometry played an important role in the formulation of the general theory of relativity by A. Einstein, and, further, its development was related to the creation of the apparatus of tensor analysis. Riemannian geometry and its many generalizations have been successfully developed, particularly in that part known as [[Riemannian geometry in the large|Riemannian geometry in the large]], and find wide and profound application in mechanics and physics [[#References|[4]]].

The basic concepts of Riemannian geometry are the following.

==Scalar product.==
In each tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r0821509.png" />, the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215010.png" /> determines a scalar (inner) product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215011.png" /> according to the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215012.png" /></td> </tr></table>

The converse is also true: If for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215014.png" /> a scalar product is defined which depends differentiably on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215015.png" />, then it defines a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215016.png" /> with the properties listed above. The degrees of smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215018.png" /> vary, depending on the problem posed. In most cases it is sufficient to demand that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215019.png" /> be three times continuously differentiable and that the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215020.png" /> be twice continuously differentiable (below, the necessary degree of smoothness will not be indicated). In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215021.png" /> with a local basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215023.png" />, the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215024.png" /> take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215025.png" /></td> </tr></table>

so that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215026.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215027.png" /></td> </tr></table>

==A Riemannian space as a metric space.==
The length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215028.png" /> of a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215029.png" /> is determined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215031.png" /> is the tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215032.png" />. The length of a piecewise-smooth curve is equal to the total length of its smooth parts. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215033.png" /> is the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215034.png" /> in local coordinates, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215035.png" /></td> </tr></table>

In view of this formula, the metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215036.png" /> is written in the conventional form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215037.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215038.png" /> is called the element of length, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215039.png" /> being the coefficients of the metric (first fundamental) form. The angle between two curves at a point of intersection is defined as the angle between the tangents to them. The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215040.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215041.png" /> which belongs to a coordinate neighbourhood is determined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215042.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215043.png" />. The volume of an arbitrary domain is equal to the sum of the volumes of its parts, each of them lying in a specific coordinate neighbourhood.

The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215044.png" /> between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215045.png" /> is defined as the greatest lower bound of the lengths of all piecewise-smooth curves that join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215047.png" />. The metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215048.png" /> in an arbitrary connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215049.png" /> is defined in the same way. Two Riemannian spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215051.png" /> are called isometric if there is a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215052.png" /> under which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215053.png" /></td> </tr></table>

or, which is the same, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215055.png" /> is an arbitrary curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215057.png" /> is an isometry, then for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215058.png" /> there is a coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215059.png" /> and a coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215060.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215063.png" />. An isometric mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215064.png" /> onto itself is called a motion.

A curve with ends at two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215066.png" /> is called a shortest curve if its length is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215067.png" />. A stationary curve of the length functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215068.png" /> is called a geodesic. Each shortest curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215069.png" /> is a geodesic and each sufficiently small arc of a geodesic is a shortest curve. A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215070.png" /> is called geodesically convex if the shortest curves, determined from the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215071.png" />, are geodesics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215072.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215074.png" />, are the equations of a geodesic in a local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215075.png" />, then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215076.png" /> satisfy a system of equations that, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215077.png" /> is a parameter proportional to the arc length, takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215078.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215079.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215080.png" /></td> </tr></table>

are the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215081.png" /> are the elements of the matrix inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215083.png" />.

A Riemannian space is called complete (geodesically complete) if it is complete as a metric space (if any arc of a geodesic can be extended indefinitely on both sides). A Riemannian space is complete if and only if it is geodesically complete. In a complete Riemannian space any two points can be connected by a shortest curve (that is not necessarily unique). The structure of a complete Riemannian space may be introduced on any differentiable manifold.

==A Riemannian space as a manifold with a connection.==
A [[Covariant derivative|covariant derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215084.png" /> is called symmetric and compatible with the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215085.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215086.png" /> if the symmetry conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215087.png" /></td> </tr></table>

and compatibility conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215088.png" /></td> </tr></table>

are fulfilled, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215089.png" /> are vector fields and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215090.png" /> is their Lie bracket. These conditions determine the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215091.png" /> uniquely in terms of the field of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215092.png" />. In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215093.png" /> the components of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215094.png" /> take the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215095.png" />, coincide with the Christoffel symbol of the first kind, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215096.png" /></td> </tr></table>

The covariant derivative of any tensor is determined by an analogous formula.

A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215097.png" /> along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215098.png" /> is called parallel if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215099.png" />. Analytically a parallel field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150100.png" /> is determined by a solution to the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150101.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150102.png" /> are the equations of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150103.png" />. The solutions to this system under different initial conditions determine a transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150104.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150105.png" />; it turns out to be an isometry and is called the Levi-Civita parallel displacement. The result of the transfer depends, as a rule, not only on the end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150107.png" />, but also on the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150108.png" /> itself. A curve for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150109.png" /> is a geodesic, and this property of a geodesic can be taken as its definition.

==Submanifolds of a Riemannian space.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150111.png" />, is a differentiable submanifold of a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150112.png" />, then in each tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150113.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150114.png" /> a scalar product is induced, and thus there arises on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150115.png" /> the structure of a Riemannian space with a metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150116.png" />, the components of which are calculated by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150117.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150118.png" /> are the equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150119.png" /> in local coordinates. The extrinsic geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150121.png" />, is described by the second fundamental forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150122.png" />, which are determined for each unit normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150123.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150124.png" /> by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150125.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150127.png" /> are tangent vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150128.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150129.png" /> is an arbitrary field of unit normals containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150130.png" />. For any form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150131.png" /> the normal curvatures, the principal directions and curvatures, the [[Mean curvature|mean curvature]], the [[Complete curvature|complete curvature]], etc., are defined, and the Gauss–Codazzi–Ricci equations can be derived, connecting the coefficients of the first and second fundamental forms. Important classes of submanifolds are characterized by properties of the second fundamental form, i.e. minimal, totally geodesic, convex, etc. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150132.png" /> (a smooth curve) a theory has been constructed similar to that of curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150133.png" />, the first, second, etc., curvatures have been determined, and equations similar to the [[Frénet formulas|Frénet formulas]] have been derived. The first curvature of a curve, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150134.png" />, is generally called the geodesic curvature, and is calculated from the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150135.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150136.png" /> is the arc length parameter; in local coordinates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150137.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150138.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150139.png" /> are the equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150140.png" />.

Many problems of Riemannian geometry are connected with [[Isometric immersion|isometric immersion]] of one Riemannian space into another, and with the study of the properties of such immersions. These problems are difficult and little research has been done on them (in the two-dimensional case they were studied in more detail).

The exponential mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150141.png" /> is determined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150142.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150143.png" /> is the end of the arc of the geodesic that starts at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150144.png" /> with direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150145.png" /> and length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150146.png" />. If a coordinate system is introduced into a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150147.png" /> by associating with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150148.png" /> the Cartesian coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150149.png" />, then it turns out that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150150.png" /></td> </tr></table>

that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150151.png" /></td> </tr></table>

these are the so-called (Riemannian) normal coordinates.

==Curvature.==
If normal coordinates are introduced in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150152.png" />, then the components of the metric tensor take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150153.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150154.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150156.png" />. From this, an important property of a Riemannian metric can be derived: For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150157.png" /> the exponential mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150158.png" /> possesses the property

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150159.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150160.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150161.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150162.png" />. In general it is impossible to obtain a higher-order coincidence of the metrics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150164.png" /> by a more successful choice of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150165.png" />. Therefore the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150166.png" /> characterize the deviation between the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150167.png" /> and the Euclidean metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150168.png" />. These coefficients are components of the so-called curvature tensor or Riemann–Christoffel tensor (at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150169.png" />). In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150170.png" /> they are expressed in terms of the coefficients of the metric tensor and its first and second derivatives by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150171.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150172.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150173.png" /></td> </tr></table>

A series of other concepts is associated with the curvature tensor. These also (from various sides) characterize the extent of the deviation of the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150174.png" /> from the Euclidean metric. Thus, in terms of the curvature tensor one can define the Ricci tensor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150175.png" /></td> </tr></table>

and the Einstein tensor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150176.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150177.png" /></td> </tr></table>

is known as the scalar curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150178.png" />.

The trilinear transformation that assigns the field

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150179.png" /></td> </tr></table>

to the three vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150180.png" />, is called the curvature transformation. Its properties are:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150181.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150182.png" /> (the first Bianchi identity);

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150183.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150184.png" />.

In addition, the second Bianchi identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150185.png" /></td> </tr></table>

holds. The curvature transformation may be connected by some construction with the parallel transfer. The algebraic properties of the curvature tensor are derived from the properties of the curvature transformation, since in terms of it, and in particular in terms of the  "bi-quadratic form"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150186.png" />, the curvature tensor (or, more precisely, its value on the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150187.png" />) can be uniquely (algebraically) expressed (see [[Curvature|Curvature]]).

==Sectional curvature.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150188.png" /> be a two-dimensional surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150189.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150190.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150191.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150192.png" /> be a simply-closed curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150193.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150194.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150195.png" /> be the area of the domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150196.png" /> bounded by the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150197.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150198.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150199.png" /> be the vector obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150200.png" /> by parallel transfer along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150201.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150202.png" /> be the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150203.png" /> and the tangential component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150204.png" />. Then when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150205.png" /> is contracted to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150206.png" />, the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150207.png" /> exists and is called the Riemannian sectional curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150208.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150209.png" /> in the given two-dimensional direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150210.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150211.png" /> does not depend on the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150212.png" /> but only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150213.png" />). The sectional curvature shows the extent of  "bending"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150214.png" /> at a given point and in a given two-dimensional direction. In general, this bending varies with different two-dimensional directions; if, however, at each point the curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150215.png" /> does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150216.png" />, then it does not change from point to point (Schur's theorem). The identical vanishing of the sectional curvature is a necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150217.png" /> to be locally isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150218.png" /> (in the large it can be different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150219.png" />). The sectional curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150220.png" /> may be connected with other objects of Riemannian geometry as well, such as the defect (excess) of a geodesic triangle (see [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]]). Riemann defined the sectional curvature as the Gaussian curvature of the two-dimensional surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150221.png" /> calculated by Gauss' formula at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150222.png" />. The metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150223.png" /> is uniquely defined by the sectional curvature in the following sense: If the sectional curvatures of two manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150224.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150225.png" /> are constant and equal to the same number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150226.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150227.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150228.png" /> are locally isometric, and if they are also both simply connected, then they are simply isometric. A simply-connected Riemannian space of constant sectional curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150229.png" /> is isometric to: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150230.png" />-dimensional [[Lobachevskii space|Lobachevskii space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150231.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150232.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150233.png" />-dimensional [[Euclidean space|Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150234.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150235.png" />; and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150236.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150237.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150238.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150239.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150240.png" />. In general, the following result is known: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150241.png" /> is an analytic Riemannian space of non-constant sectional curvature and if there is a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150242.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150243.png" />, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150244.png" /> the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150245.png" /> is an isometry (cf. [[Isometric mapping|Isometric mapping]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150246.png" /> this statement has been proved under certain additional assumptions, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150247.png" /> the theorem is not true. However, it is not known (1983), apart from the two-dimensional case, what the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150248.png" /> must be in order for it to have a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150249.png" /> for which it is the sectional curvature. Only some negative results have so far been obtained in this direction.

The sectional curvature is connected with the curvature transformation by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150250.png" /></td> </tr></table>

and in terms of the components of the curvature tensor it is expressed as:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150251.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150252.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150253.png" /> is determined by the vectors

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150254.png" /></td> </tr></table>

The value of the Ricci tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150255.png" /> at a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150256.png" /> is connected with the sectional curvature in the following way: Let the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150257.png" /> form an orthonormal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150258.png" />; then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150259.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150260.png" /> is the two-dimensional direction of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150261.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150262.png" />.

==Special classes of Riemannian spaces.==
In addition to general (arbitrary) Riemannian spaces there are Riemannian spaces onto which additional structures may be introduced. These structures arise when some kind of geometric or algebraic conditions are directly imposed on the metrics. This is how some important classes of Riemannian spaces are defined: a manifold of constant sectional curvature (see [[Space forms|Space forms]]), a [[Homogeneous space|homogeneous space]], a [[Symmetric space|symmetric space]], Hermitian and Kähler manifolds (cf. [[Kähler manifold|Kähler manifold]]), Einstein spaces, etc.

==Generalizations.==
The development of the ideas of Riemannian geometry and geometry in the large has led to a series of generalizations of the concept of Riemannian geometry.

Pseudo-Riemannian geometry is the theory of a [[Pseudo-Riemannian space|pseudo-Riemannian space]]. This is a differentiable manifold on which a non-degenerate symmetric tensor field is given.

Finsler geometry is the theory of a differentiable manifold in the tangent bundle of which a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150263.png" /> is given that is homogeneous of the first degree in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150264.png" />. The length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150265.png" /> of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150266.png" /> is calculated as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150267.png" /></td> </tr></table>

Spaces of bounded curvature refers to the theory of two-dimensional metric manifolds with an [[Internal metric|internal metric]] (without any assumption of smoothness) on which the total curvature of any bounded Borel set is defined. Related to this theory is the intrinsic geometry of convex surfaces [[#References|[5]]]. This class of metric spaces may be obtained by adjoining to two-dimensional Riemannian spaces two-dimensional metric manifolds whose metric in a neighbourhood of each point admits a uniform approximation by Riemannian metrics with integrals of the absolute Gaussian curvature that are bounded in aggregate.

Spaces of curvature not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150269.png" /> refers to the theory of complete metric manifolds with internal metrics, in which the sum of the interior angles of triangles which are made up of shortest curves does not exceed the sum of the angles of a triangle in a plane of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150270.png" /> with sides of the same length (furthermore, it is assumed that any two points may be connected by a single shortest curve). See also [[Geodesic geometry|Geodesic geometry]]; [[Conformal geometry|Conformal geometry]]; [[Riemannian space, generalized|Riemannian space, generalized]]; [[Riemannian geometry in the large|Riemannian geometry in the large]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,   "Über die Hypothesen, welche der Geometrie zu Grunde liegen" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.P. Eisenhart,   "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Gromoll,   W. Klingenberg,   W. Meyer,   "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.D. Aleksandrov,   "Die innere Geometrie der konvexen Flächen" , Akademie Verlag  (1955)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.D. Burago,   V.A. Zalgaller,   "Convex sets in Riemannian spaces of non-negative curvature"  ''Russian Math. Surveys'' , '''32''' :  3  (1977)  pp. 1–57  ''Uspekhi Mat. Nauk'' , '''32''' :  3  (1977)  pp. 3–55</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.W. Milnor,   "Morse theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Cartan,   "Leçons sur la géométrie des espaces de Riemann" , Gauthier-Villars  (1928)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R.S. Kulkarni,   "Curvature and metric"  ''Ann. of Math.'' , '''91''' :  2  (1970)  pp. 311–331</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.A. Wolf,   "Spaces of constant curvature" , Publish or Perish  (1977)</TD></TR></table>



====Comments====
Next to the notion of  "geodesically convex" , in the West one also uses the following. A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150271.png" /> is called: 1) simple if for every pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150272.png" /> there is at most one joining geodesic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150273.png" />; 2) convex if for every pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082150/r082150274.png" /> there is exactly one joining shortest curve; and 3) strongly convex if it is simple and convex.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. O'Neill,   "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Gallot,   D. Hulin,   J. Lafontaine,   "Riemannian geometry" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.M. Boothby,   "An introduction to differentiable manifolds and Riemannian geometry" , Acad. Press  (1975)</TD></TR></table>