Geometry in the large

global geometry

Geometric theories dealing with the entire geometric object (the entire curve, the entire surface, the entire space; in a similar manner, the entire vector field, the entire tensor field or the entire field of some other objects; in a similar manner, the entire mapping of a geometric figure or of a field of geometric objects into another). The term "geometry in the large" (in German: "Geometrie im Grossen" ) appeared in the German mathematical literature at the beginning of the 20th century. It differs from local geometry, in which a geometric object (field, mapping) is studied in sufficiently small domains only, as is the case in classical differential geometry, the methods of which proved to be inadequate in geometry in the large. In the absence of this difference (e.g. in elementary geometry, in the topology of manifolds) the term "geometry in the large" is not employed.

The qualitative difference between properties "in the large" and properties "in the small" is manifested, first and foremost, in problems concerned with rigidity, deformation and isometric immersion of surfaces (e.g. a small piece of a convex surface can be bent preserving its convexity, while this is no longer possible for the entire surface of a convex body), in the behaviour of geodesic lines (e.g. in a small domain two points on a smooth surface are connected by a unique geodesic, while they are connected by an infinite number of geodesics on an entire closed surface), the possibility of specifying a metric with given properties on different manifolds (e.g. a metric of everywhere-positive curvature can be specified on a complete surface if the latter is homeomorphic to a sphere, a plane or a projective plane only). Such problems generated independent theories, such as variational calculus in the large. The development of modern methods of differential geometry better suited to the study of geometry in the large yielded numerous qualitative results, as well as quantitative relations in the large, for regular geometric structures on multi-dimensional singularity-free manifolds.

Singularities are often inevitably generated when smooth immersed manifolds or fields on such manifolds are extended. In addition, the solutions of many extremal problems are obtained on irregular objects. For this reason many problems of geometry in the large are more naturally posed in classes comprising irregular objects. Methods other than those of differential geometry are needed for this purpose. Such approaches, combining studies in the large with studies of singularities, were developed for two-dimensional surfaces by the geometric school of A.D. Aleksandrov, N.V. Efimov and A.V. Pogorelov, who in fact obtained the most advanced results in the theory of surfaces.

References

[1]	S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)
[2]	A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)
[3]	N.V. Efimov, "Geometry "in the large" " , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) (In Russian)
[4]	A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)
[5]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2]	J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975)
[a3]	S.-T. Yau, "Problem section" S.-T. Yau (ed.) , Seminar on differential geometry , Ann. Math. Studies , 102 , Princeton Univ. Press (1982) pp. 669–706
[a4]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)