Riemannian geometry in the large

The branch of Riemannian geometry that examines the connections between the local and global characteristics of Riemannian manifolds. The term "Riemannian geometry in the large" usually refers to a specific range of problems and methods characteristic for geometry in the large. Fundamental to Riemannian geometry in the large is the study of the connection between the curvature and the topology of a Riemannian manifold. Thus, problems relating to the topological and the metric structure of Riemannian manifolds with given conditions on the curvature are studied, e.g. the question of the existence of a Riemannian metric with prescribed properties of the curvature on a given smooth manifold (the sectional curvature $ K _ \sigma $; the Ricci curvature Ric; the scalar curvature $ K _ \mathop{\rm sc} $). The major part of the results obtained relate to spaces with curvatures of constant sign. Riemannian geometry in the large is closely connected with the theory of homogeneous spaces (cf. Homogeneous space) and the variational theory of geodesics (cf. Geodesic line). For submanifolds of Riemannian manifolds, see Isometric immersion and Geometry of immersed manifolds.

The methods of Riemannian geometry in the large are of a synthetic character. In addition to local differential geometry, wide use is made of the theory of differential equations and Morse theory. The main achievements are those of the discovery of successful constructions, such as the construction of closed geodesics, of minimal films or of films of geodesics, of horospheres, and of convex sets. The study of the topology of Riemannian manifolds is usually preceded by the study of their metric properties. The latter is often accomplished by comparing a Riemannian manifold with a suitable standard space (see below: Comparison theorems).

Topological structure.

For closed surfaces the connection between the curvature and the topology is essentially determined by the Gauss–Bonnet formula (cf. Gauss–Bonnet theorem). Among the closed surfaces, only the sphere $ S ^ {2} $ and the projective plane $ P ^ {2} $ can carry a metric of positive curvature; only the torus and the Klein bottle can carry a metric of curvature zero. The structure of a Riemannian manifold of dimension $ n > 2 $ is less well known (1991). Examples of known theorems are given below.

A complete simply-connected Riemannian manifold $ M ^ {n} $ with $ K _ \sigma \leq 0 $ is diffeomorphic to $ \mathbf R ^ {n} $( the Hadamard–Cartan theorem); moreover, for any point $ x \in M ^ {n} $ the exponential mapping $ \mathop{\rm exp} _ {x} $ is a diffeomorphism of the tangent space $ T _ {x} M ^ {n} $ onto $ M ^ {n} $.

For closed Riemannian manifolds with $ K _ \sigma > 0 $, the sphere theorem is valid: A complete Riemannian manifold $ M ^ {n} $ with $ 0 < \delta \leq K _ \sigma \leq 1 $ is called $ \delta $- pinched; if it is simply connected and $ \delta > 1/4 $, then $ M ^ {n} $ is homeomorphic to $ S ^ {n} $. For even $ n $ this bound is precise. When $ \delta = 1/4 $ there are manifolds $ M ^ {n} $ that are not homeomorphic to $ S ^ {n} $: these are the symmetric spaces of rank 1 and only these (cf. Symmetric space). For odd $ n $ the theorem about the homeomorphism of $ M ^ {n} $ to $ S ^ {n} $ is true even if $ \delta = 1/4 $. When $ n < 7 $, $ n \neq 4 $, homeomorphism to $ S ^ {n} $ implies diffeomorphism. When $ n \geq 7 $, diffeomorphism to the sphere is established with a stronger pinching than in the sphere theorem (it is sufficient to take $ \delta > 0.87 $, while if $ n \rightarrow \infty $, $ \delta > 0.66 $). It is also known that with an even stronger pinching (it is sufficient to take $ \delta > 0.98 $ and $ \delta > 0.66 $ as $ n \rightarrow \infty $) a non-simply connected $ M ^ {n} $ is diffeomorphic to a space of constant curvature (the quotient space of $ S ^ {n} $ by a discrete subgroup of isometries). There are a number of results concerning the conditions on $ K _ \sigma $ that guarantee homeomorphism to a symmetric space of rank 1 (see [14], [15]).

An open, that is to say, complete non-compact, Riemannian manifold with $ K _ \sigma > 0 $ is always diffeomorphic to $ \mathbf R ^ {n} $. A set $ X \subset M ^ {n} $ is called absolutely convex if each geodesic with ends in $ X $ lies entirely in $ X $. Let $ M ^ {n} $ be an open Riemannian manifold with $ K _ \sigma \geq 0 $. Then in $ M ^ {n} $ there is a totally geodesic absolutely convex closed submanifold $ N $ such that $ M ^ {n} $ is diffeomorphic to the space $ \nu ( N) $ of the normal bundle of $ N $ in $ M ^ {n} $( if $ K _ \sigma > 0 $, then $ \mathop{\rm dim} N = 0 $). When $ \mathop{\rm dim} N = 1 $ or $ n- 1 $, and always for homogeneous spaces, $ M ^ {n} $ is even isometric to $ \nu ( N) $ with the standard metric of the normal bundle. When $ n \leq 3 $, this gives a complete classification of open Riemannian manifolds with $ K _ \sigma \geq 0 $.

A straight line is a complete geodesic that is a shortest curve on any segment of it. The cylinder theorem states: An open $ M ^ {n} $ with $ K _ \sigma \geq 0 $ is isometric to the direct metric product $ M ^ {n-} k \times \mathbf R ^ {k} $, $ 0 \leq k \leq n $, where $ M ^ {n-} k $ does not contain straight lines. The condition $ K _ \sigma \geq 0 $ can here be replaced by $ \mathop{\rm Ric} \geq 0 $.

Fundamental group.

When $ K _ \sigma > 0 $ and $ n $ is even, a closed $ M ^ {n} $ is either orientable and simply connected or non-orientable and the fundamental group $ \pi _ {1} ( M ^ {n} ) = \mathbf Z _ {2} $; when $ n $ is odd, it is always orientable but little is known about $ \pi _ {1} ( M ^ {n} ) $ beyond the limits of the sphere theorem. Even for $ M ^ {n} $ of constant curvature $ K _ \sigma = 1 $, the full description of the possible structures of $ \pi _ {1} ( M ^ {n} ) $ for odd $ n $ turned out to be a difficult problem (see [9]).

If $ K _ \sigma = 0 $, then the universal covering $ {\widetilde{M} } {} ^ {n} $ of $ M ^ {n} $ is isometric to $ \mathbf R ^ {n} $ and the fundamental group $ \pi _ {1} ( M ^ {n} ) $ is isomorphic to a discrete group of isometries of $ \mathbf R ^ {n} $ without fixed points; it contains the subgroup of translations as a subgroup of finite index. (Thus, $ M ^ {n} $ permits a finite isometric covering by a flat torus.)

If $ K _ \sigma \leq 0 $ on $ \widetilde{M} {} ^ {n} $, then $ {\widetilde{M} } {} ^ {n} $ is diffeomorphic to $ \mathbf R ^ {n} $. Therefore all homotopy groups $ \pi _ {i} ( M ^ {n} ) $ for $ i > 1 $ are trivial and the homotopy type is determined by $ \pi _ {1} ( M ^ {n} ) $. If $ K _ \sigma < 0 $, then $ \pi _ {1} ( M ^ {n} ) $ is completely non-commutative in the sense that any of its Abelian (and even any of its solvable) subgroups is an infinite cyclic group. When $ K _ \sigma \leq 0 $, the following is known. Let $ \Gamma $ be a solvable subgroup of $ \pi _ {1} ( M ^ {n} ) $. Then $ \Gamma $ is isomorphic to a discrete group of isometries of $ \mathbf R ^ {n} $( without fixed points) and $ M ^ {n} $ contains a compact totally geodesic submanifold which is isometric to $ \mathbf R ^ {n} / \Gamma $. Instead of $ K _ \sigma \leq 0 $ it is sufficient in this case to require the absence of conjugate points on the geodesics.

For two manifolds of the same constant negative curvature and the same dimension $ n \geq 3 $, isomorphism of their $ \pi _ {1} $' s implies isometry (Mostow's rigidity theorem).

Riemannian manifolds for which $ \max | K _ \sigma | \cdot \mathop{\rm diam} M ^ {n} < \epsilon $ are called $ \epsilon $- flat. For an arbitrary $ \epsilon > 0 $ such manifolds can be topologically different from locally flat manifolds. For them, for any $ n \geq 2 $, there is an $ \epsilon _ {n} $ such that for an $ \epsilon _ {n} $- flat $ M ^ {n} $ there is a nilpotent subgroup of finite index in $ \pi _ {1} ( M ^ {n} ) $. In this case $ M ^ {n} $ permits a finite (with multiplicity dependent only on $ M $) covering diffeomorphic to the quotient space of a nilpotent Lie group by a discrete subgroup of it (see [8]).

A complete Riemannian manifold with curvature $ \mathop{\rm Ric} \geq a > 0 $ has finite $ \mathop{\rm diam} M ^ {n} \leq \pi / \sqrt a $ and thus a finite group $ \pi _ {1} ( M ^ {n} ) $. If $ \mathop{\rm Ric} \geq 0 $ for a closed $ M ^ {n} $, there is a finite normal subgroup $ \Gamma \subset \pi _ {1} ( M ^ {n} ) $ such that $ \pi _ {1} / \Gamma $ is a discrete group of isometries of $ \mathbf R ^ {k} $, $ 0 \leq k \leq n $; moreover, $ {\widetilde{M} } {} ^ {n} $ decomposes into the direct metric product $ M ^ {*} \times \mathbf R ^ {k} $, where $ M ^ {*} $ is closed, the decomposition is invariant relative to $ \pi _ {1} ( M ^ {n} ) $ and $ \Gamma $ is trivial in $ \mathbf R ^ {k} $.

In addition to the study of $ \pi _ {1} ( M ^ {n} ) $, several estimates of the Betti numbers $ b _ {k} $ have been made using the theory of harmonic differential forms for $ \delta $- pinched $ M ^ {n} $. Thus, $ b _ {2} = 0 $ when $ \delta > ( n- 3)( 4n- 9) ^ {-} 1 $ and $ n \geq 5 $ is odd.

Comparison theorems.

Many global properties of Riemannian manifolds are proved by comparing the structures in the Riemannian manifolds under consideration to similar structures on a standard space. This usually is a manifold of constant curvature, or, more rarely, another symmetric space. Below, a $ c $- plane is $ \mathbf R ^ {2} $ when $ c = 0 $, the sphere $ S _ {c} ^ {2} $ of radius $ c ^ {-} 1/2 $ when $ c > 0 $ and the Lobachevskii plane of curvature $ c $ when $ c < 0 $.

The Toponogov theorem on the comparison of angles has many applications: In a Riemannian manifold $ M ^ {n} $, let all $ K _ \sigma \geq c $, let $ \alpha _ {1} , \alpha _ {2} , \alpha _ {3} $ be the angles of a triangle made of shortest curves, and let $ \alpha _ {1} ^ \prime , \alpha _ {2} ^ \prime , \alpha _ {3} ^ \prime $ be the corresponding angles of a triangle with sides of the same length in the $ c $- plane; then $ \alpha _ {i} \geq \alpha _ {i} ^ \prime $. If $ K _ \sigma \leq c $ and if any two points of the sides of the triangle in $ M ^ {n} $ in question can be joined by a single shortest curve, then $ \alpha _ {i} \leq \alpha _ {i} ^ \prime $. This theorem is equivalent to the following convexity condition: If in $ M ^ {n} $ the shortest curves $ ab, ac $ form the same angle as the shortest curves $ a ^ \prime b ^ \prime , a ^ \prime c ^ \prime $ of the same length in the $ c $- plane, then $ \rho _ {M ^ {n} } ( b, c) \leq \rho _ {c} ( b ^ \prime , c ^ \prime ) $. What is being compared here is essentially the rate of divergence of the shortest curves.

Rauch's comparison theorem compares the rate of movement of the end points $ b $ and $ b ^ \prime $ of two shortest curves $ ab $, $ a ^ \prime b ^ \prime $ in two Riemannian manifolds $ M ^ {n} $ and $ M ^ \prime n $ when $ ab $ and $ a ^ \prime b ^ \prime $ turn around their origins $ a $, $ a ^ \prime $ with the same rate, under conditions when (in a natural comparison) the sectional curvatures in $ M ^ {n} $ are not less than in $ M ^ \prime n $. Then the rate of movement of $ b $ is not greater than the rate of $ b ^ \prime $. In the fundamental case (comparison with a $ c $- plane), Rauch's theorem is equivalent to the infinitesimal version of the theorem on the comparison of angles.

There are theorems similar to Rauch's in which the points $ a $, $ a ^ \prime $ move on hypersurfaces to which $ ab $, $ a ^ \prime b ^ \prime $ remain orthogonal. There are also comparison theorems for volumes of tubular neighbourhoods of submanifolds (see [13], [16]).

Extremal theorems.

Comparison theorems lead to estimates of such characteristics of $ M ^ {n} $ as the diameter, the radius of injectivity, the length of a closed geodesic, the volume of a sphere of given radius, etc. Extremal theorems give answers to questions concerning cases of achieving equality in such estimates.

For $ M ^ {n} $ with $ K _ \sigma \geq 1 $ one always has $ \mathop{\rm diam} M ^ {n} \leq \pi $. Equality is achieved only for the unit sphere. If $ M ^ {n} $ is closed and $ 0 < K _ \sigma \leq 1 $ when $ n $ is even or $ 1/4 < K _ \sigma \leq 1 $ when $ n $ is odd, then the radius of injectivity $ r _ \mathop{\rm in} ( M ^ {n} ) \geq \pi $ and the length of a closed geodesic $ \geq 2 \pi $. If in this case there is a closed geodesic $ \gamma $ of length $ 2 \pi $ in $ M ^ {n} $, then, if $ n $ is even, there is a totally geodesic surface in $ M ^ {n} $ which contains $ \gamma $ and which is isometric to $ S _ {1} ^ {n} $, while if $ 1/4 < K _ \sigma \leq 1 $, independently of the parity of $ n $, $ M ^ {n} $ is isometric to $ S _ {1} ^ {n} $( see [6]). The volume of a sphere $ D $ of radius $ r < r _ \mathop{\rm in} ( M ^ {n} ) $ in an $ M ^ {n} $ with $ K _ \sigma \leq c $ $ ( K _ \sigma \geq c) $ is not less (not greater) than the volume of the sphere $ D _ {c} $ of the same radius in the space of constant curvature $ c $, with equality only if $ D $ is isometric to $ D _ {c} $.

Extremal theorems are not always connected with estimates of the curvature. For example, for any point in a closed surface $ F $, let the set of points conjugate to it consist of a single point. Then $ F $ is isometric to a sphere.

Finiteness of topological types.

Among the closed Riemannian manifolds with uniformly bounded curvatures and radii of injectivity bounded from below, $ \mathop{\rm Vol} M ^ {n} < C _ {1} $, $ r _ \mathop{\rm in} > C _ {2} > 0 $, $ K _ \sigma < C _ {3} $, only finitely many are homotopically pairwise non-equivalent, and if $ K _ \sigma < C _ {3} $ is changed into $ | K _ \sigma | < C _ {3} $, only finitely many of them are pairwise non-homeomorphic. In this statement the condition $ r _ \mathop{\rm in} > C _ {2} > 0 $ can be replaced by the conditions $ \mathop{\rm Vol} M ^ {n} \geq C _ {4} > 0 $, $ \mathop{\rm diam} M ^ {n} < C _ {5} $, which imply it but are more easily verifiable (see [14]).

For a Riemannian manifold with $ K _ \sigma $ of fixed sign, the conditions that guarantee the finiteness of its topological type are simplified. For example, for even $ n $ and $ K _ \sigma > 0 $ the condition $ \max K _ \sigma < C \min K _ \sigma $ is sufficient.

When $ n > 3 $, for $ M ^ {n} $ with $ - 1 \leq K _ \sigma < 0 $ the estimate $ \mathop{\rm Vol} M ^ {n} > C ( 1 + \mathop{\rm diam} M ^ {n} ) $ is true. Therefore, when $ n \neq 3 $, the number of topological types of closed Riemannian manifolds that satisfy the conditions $ \mathop{\rm Vol} M ^ {n} < C _ {1} $, $ C _ {2} < K _ \sigma < 0 $, is finite. But when $ n = 3 $, there is an infinite number of pairwise non-homeomorphic $ M ^ {3} $ satisfying these conditions (see [12]).

Metrics with prescribed curvature.

Let $ \chi $ be the Euler characteristic of a closed surface $ M ^ {2} $. In order that a smooth function $ f $ on $ M ^ {n} $ be the curvature of a Riemannian metric in $ M ^ {2} $ it is necessary that $ \max f > 0 $ when $ \chi > 0 $, $ \min f < 0 $ when $ \chi < 0 $ and $ f $ changes sign or $ f \equiv 0 $ when $ \chi = 0 $. These conditions are also sufficient. The condition $ \int _ {M ^ {2} } \omega = 2 \pi \chi $ is necessary and sufficient for the $ 2 $- form $ \omega $ to be the curvature form $ \omega = \int K dS $ of a Riemannian metric in $ M ^ {2} $. If $ M ^ {2} $ is an open submanifold of a closed manifold $ N ^ {2} $, then any smooth $ f $ on $ M ^ {2} $ is the curvature of a (possibly incomplete) Riemannian metric in $ M ^ {2} $. Necessary and sufficient conditions under which $ f $ is the curvature of a complete Riemannian metric in a non-compact surface have been ascertained for finitely-connected surfaces.

As the dimension increases, the number of independent components of the curvature tensor increases faster than the number of components of the metric tensor. The conditions under which the given tensor field is, at least locally, the field of the curvature tensor of a certain metric are unknown (1991). But for the scalar curvature, when $ n > 2 $, each smooth function $ f $ on a closed $ M ^ {n} $ for which $ \min f < 0 $ is the scalar curvature of a Riemannian metric in $ M ^ {n} $( see [4]). There are manifolds which do not permit a metric with a positive scalar curvature, as is the case with a three-dimensional torus (see [5]).

Convex functions.

The existence of a scalar function $ f $ on a Riemannian manifold $ M ^ {n} $ that is convex along any geodesic imposes strict limitations on the structure of such an $ M ^ {n} $. For example, if there is a convex function $ f $ on $ M ^ {n} $, then $ \mathop{\rm Vol} M ^ {n} = \infty $. If $ f $ is strictly convex and if for any $ c $ the sets $ f ^ { - 1 } ( c) $ are compact, then $ M ^ {n} $ is diffeomorphic to $ \mathbf R ^ {n} $.

Convex functions can be constructed in a number of cases. For example, when $ K _ \sigma \leq 0 $, the functions $ f _ {1} ( x) = \rho ( x, p) $, $ f _ {2} ( x) = \rho ( x, p) ^ {2} $, $ p \in M ^ {n} $, are convex. If $ K _ \sigma \leq 0 $ and $ \gamma : M ^ {n} \rightarrow M ^ {n} $ is an isometry, then the function $ \delta _ \gamma ( x) = \rho ( x, \gamma x) $ is convex. When $ K _ \sigma \geq 0 $, there are convex $ f $ with compact $ f ^ { - 1 } ( c) $; this is connected with the absolute convexity (when $ K _ \sigma \geq 0 $) of the complements to horospheres and with the fact that if $ K _ \sigma \geq 0 $, then the convexity of a set $ U \subset M ^ {n} $ implies the convexity of the set $ \{ {x \in U } : {\rho ( x, \partial U) \geq t } \} $.

Problems of Riemannian geometry in the large have also been studied for Riemannian manifolds with additional structures, such as for Kähler manifolds (see [10]).

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	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
[3]	J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975)
[4]	, Research on the metric theory of surfaces , Moscow (1980) (In Russian; translated from English and French)
[5]	R. Schoen, S.-T. Yau, "Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with non-negative scalar curvature" Ann. of Math. , 110 (1979) pp. 127–142
[6]	V.A. Toponogov, "Extremal theorems for Riemann spaces with curvature bounded above I" Sib. Math. J. , 15 : 6 (1974) pp. 954–971 Sibirsk. Mat. Zh. , 15 : 6 (1974) pp. 1348–1371
[7]	M. Gromov, H.B., jr. Lawson, "Spin and scalar curvature in the presence of a fundamental group I" Ann. of Math. , 111 : 2 (1980) pp. 209–230
[8]	P. Buser, H. Karcher, "Gromov's almost flat manifolds" Astérique , 81 (1981)
[9]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1977)
[10]	S.I. Goldberg, "Curvature and homology" , Acad. Press (1962)
[11]	A.L. Besse, "Manifolds all of whose geodesics are closed" , Springer (1978)
[12]	W. Thurston, "The geometry and topology of 3-manifolds" , Princeton Univ. Press (1978) (Preprint)
[13]	E. Heintze, H. Karcher, "A general comparison theorem with applications to volume estimates for submanifolds" Ann. Sci. Ecole Norm. Sup. , 11 : 4 (1978) pp. 451–470
[14]	J. Cheeger, "Pinching theorems for a certain class of Riemannian manifolds" Amer. J. Math. , 91 : 3 (1969) pp. 807–834
[15]	Min-Do, E. Ruh, "Comparison theorems for compact symmetric spaces" Ann. Sci. Ecole Norm. Sup. , 12 (1979) pp. 335–353
[16]	A. Gray, "Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula" Topology , 21 : 2 (1982) pp. 201–228

Comments

For some more results on the interrelations between curvature and fundamental group cf. (the editorial comments to) Polycyclic group and Polynomial and exponential growth in groups and algebras.

Let $ \xi $ be a unit length tangent vector at $ m \in M $. Let $ \gamma _ \xi $ be its geodesic. For sufficiently small $ t $ the distance between the starting point $ m = \gamma _ \xi ( 0) $ and $ \gamma _ \xi ( t) $ is $ t $, but this may fail for larger $ t $. The cut value function (on the sphere subbundle of the tangent bundle) is defined by

$$

\mathop{\rm Cutval} ( \xi )  =  t _ {0}  \iff \

The cut locus of $ m \in M $ is by definition the set of points $ \{ \mathop{\rm exp} _ {m} ( \mathop{\rm Cutval} ( \xi ) \xi ) \} = \mathop{\rm Cutloc} ( m) $, where $ \xi $ runs over all unit length tangent vectors at $ m $ and $ \mathop{\rm exp} _ {m} : T _ {m} M \rightarrow M $ is the exponential mapping at $ m \in M $; cf. Exponential mapping and the subsection "Exponential mapping" in Riemannian geometry.

The radius of injectivity, $ \mathop{\rm Inj} ( m) $, at $ m \in M $ is defined as $ \inf \{ { \mathop{\rm Cutval} ( \xi ) } : {\xi \in T _ {m} M, \| \xi \| = 1 } \} $, and the global radius of injectivity of $ M $ is $ \inf \{ { \mathop{\rm Inj} ( m ) } : {m \in M } \} $. It is always positive if $ M $ is compact, but can be zero if $ M $ is non-compact.

The cut locus was introduced by H. Poincaré under the name "ligne de partage" . Every compact surface admits a Riemannian metric of constant curvature. "Cutting up" a higher-genus surface $ M $ along a cut locus yields a particularly nice representation of $ M $ as a domain bounded by a polygon on which edges are pairwise identified, [a3]. For the sphere, the genus 0 case, the cut locus of a point is the single diametrically-opposite point.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. Sect. 11.4 (Translated from French)
[a2]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) pp. Chapt. 2 (Translated from German)
[a3]	S. Gallot, D. Hulin, J. Lafontaine, "Riemannian geometry" , Springer (1987) (Translated from French)
[a4]	W.M. Boothby, "An introduction to differentiable manifolds and Riemannian geometry" , Acad. Press (1975)
[a5]	W. Greub, S. Halperin, R. Vanstone, "Connections, curvature, and cohomology" , 1–3 , Acad. Press (1972–1976)