# Riemannian geometry

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The theory of Riemannian spaces. A Riemannian space is an -dimensional connected differentiable manifold on which a differentiable tensor field of rank 2 is given which is covariant, symmetric and positive definite. The tensor is called a metric tensor. Riemannian geometry is a multi-dimensional generalization of the intrinsic geometry (cf. Interior geometry) of two-dimensional surfaces in the Euclidean space . 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 of a surface in .

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 -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" [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, and find wide and profound application in mechanics and physics [4].

The basic concepts of Riemannian geometry are the following.

## Scalar product.

In each tangent space , , the tensor determines a scalar (inner) product according to the formula

The converse is also true: If for any in a scalar product is defined which depends differentiably on , then it defines a tensor field with the properties listed above. The degrees of smoothness of and vary, depending on the problem posed. In most cases it is sufficient to demand that be three times continuously differentiable and that the field of be twice continuously differentiable (below, the necessary degree of smoothness will not be indicated). In local coordinates with a local basis , , the components of take the form

so that

where

## A Riemannian space as a metric space.

The length of a smooth curve is determined by the formula

where is the tangent vector to . The length of a piecewise-smooth curve is equal to the total length of its smooth parts. If is the equation of in local coordinates, then

In view of this formula, the metric in is written in the conventional form

and is called the element of length, the functions 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 of a domain which belongs to a coordinate neighbourhood is determined by the formula

where . 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 between two points is defined as the greatest lower bound of the lengths of all piecewise-smooth curves that join and . The metric in an arbitrary connected domain is defined in the same way. Two Riemannian spaces and are called isometric if there is a transformation under which

or, which is the same, , where is an arbitrary curve in . If is an isometry, then for any point there is a coordinate neighbourhood and a coordinate neighbourhood such that , , . An isometric mapping of onto itself is called a motion.

A curve with ends at two points and is called a shortest curve if its length is equal to . A stationary curve of the length functional is called a geodesic. Each shortest curve in is a geodesic and each sufficiently small arc of a geodesic is a shortest curve. A domain is called geodesically convex if the shortest curves, determined from the metric , are geodesics of . If , , are the equations of a geodesic in a local coordinate system , then the functions satisfy a system of equations that, when is a parameter proportional to the arc length, takes the form

where

are the Christoffel symbols (cf. Christoffel symbol), and are the elements of the matrix inverse to , .

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 is called symmetric and compatible with the metric of the space if the symmetry conditions

and compatibility conditions

are fulfilled, where are vector fields and is their Lie bracket. These conditions determine the derivative uniquely in terms of the field of the metric tensor . In local coordinates the components of the connection take the form , coincide with the Christoffel symbol of the first kind, and

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

A vector field along a curve is called parallel if . Analytically a parallel field is determined by a solution to the system

where are the equations of the curve . The solutions to this system under different initial conditions determine a transformation of into ; 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 and , but also on the arc itself. A curve for which is a geodesic, and this property of a geodesic can be taken as its definition.

## Submanifolds of a Riemannian space.

If , , is a differentiable submanifold of a Riemannian space , then in each tangent space to a scalar product is induced, and thus there arises on the structure of a Riemannian space with a metric tensor , the components of which are calculated by the formulas

where are the equations of in local coordinates. The extrinsic geometry of , , is described by the second fundamental forms , which are determined for each unit normal to by the formula

where and are tangent vector fields on and is an arbitrary field of unit normals containing . For any form the normal curvatures, the principal directions and curvatures, the mean curvature, the 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 (a smooth curve) a theory has been constructed similar to that of curves in , the first, second, etc., curvatures have been determined, and equations similar to the Frénet formulas have been derived. The first curvature of a curve, , is generally called the geodesic curvature, and is calculated from the formula

if is the arc length parameter; in local coordinates

where

and are the equations of .

Many problems of Riemannian geometry are connected with 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 is determined by the condition where is the end of the arc of the geodesic that starts at with direction and length . If a coordinate system is introduced into a neighbourhood of a point by associating with the point the Cartesian coordinates of the point , then it turns out that

that is,

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

## Curvature.

If normal coordinates are introduced in a neighbourhood of a point , then the components of the metric tensor take the form

where as , . From this, an important property of a Riemannian metric can be derived: For any point the exponential mapping possesses the property

where as . In general it is impossible to obtain a higher-order coincidence of the metrics of and by a more successful choice of the mapping . Therefore the coefficients characterize the deviation between the metric of and the Euclidean metric of . These coefficients are components of the so-called curvature tensor or Riemann–Christoffel tensor (at the point ). In local coordinates they are expressed in terms of the coefficients of the metric tensor and its first and second derivatives by the formula

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 from the Euclidean metric. Thus, in terms of the curvature tensor one can define the Ricci tensor

and the Einstein tensor

where

is known as the scalar curvature of .

The trilinear transformation that assigns the field

to the three vector fields , is called the curvature transformation. Its properties are:

1) ;

2) (the first Bianchi identity);

3) ;

4) .

In addition, the second Bianchi identity

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" , the curvature tensor (or, more precisely, its value on the vectors ) can be uniquely (algebraically) expressed (see Curvature).

## Sectional curvature.

Let be a two-dimensional surface in passing through , let . Let be a simply-closed curve in passing through , let be the area of the domain in bounded by the curve , let , let be the vector obtained from by parallel transfer along , and let be the angle between and the tangential component of . Then when is contracted to the point , the limit exists and is called the Riemannian sectional curvature of at in the given two-dimensional direction ( does not depend on the surface but only on ). The sectional curvature shows the extent of "bending" of 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 does not depend on the choice of , 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 to be locally isometric to (in the large it can be different from ). The sectional curvature of may be connected with other objects of Riemannian geometry as well, such as the defect (excess) of a geodesic triangle (see Gauss–Bonnet theorem). Riemann defined the sectional curvature as the Gaussian curvature of the two-dimensional surface calculated by Gauss' formula at . The metric of is uniquely defined by the sectional curvature in the following sense: If the sectional curvatures of two manifolds and are constant and equal to the same number , then and 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 is isometric to: the -dimensional Lobachevskii space when ; the -dimensional Euclidean space when ; and the -dimensional sphere in of radius when . In general, the following result is known: If is an analytic Riemannian space of non-constant sectional curvature and if there is a diffeomorphism such that , then if the transformation is an isometry (cf. Isometric mapping); if this statement has been proved under certain additional assumptions, while if the theorem is not true. However, it is not known (1983), apart from the two-dimensional case, what the function must be in order for it to have a metric 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

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

where is determined by the vectors

The value of the Ricci tensor at a vector is connected with the sectional curvature in the following way: Let the vectors form an orthonormal basis in ; then

where is the two-dimensional direction of the vectors and .

## 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), a homogeneous space, a symmetric space, Hermitian and Kähler manifolds (cf. 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. 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 is given that is homogeneous of the first degree in . The length of a curve is calculated as follows:

Spaces of bounded curvature refers to the theory of two-dimensional metric manifolds with an 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 [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 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 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; Conformal geometry; Riemannian space, generalized; Riemannian geometry in the large.

#### References

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