Sunto
Questa conferenza è un rapporto su un capitolo della geometria differenziale in grande. Vengono discussi varii metodi di investigazione dei legami tra le proprietà locali della geometria differenziale riemanniana di una superficie, in particolare il segno della sua curvatura gaussiana, e la sua struttura topologica globale. Alla fine del lavoro vengono faite alcune brevi osservazioni riguardanti la possibilità di applicare metodi analoghi allo studio delle varietà n-dimensionali.
Summary
This lecture is a report upon a chapter of differential geometry in the large. We discuss various methods for investigating the connections between the local properties of the Riemann differential geometry of a surface, in particular the sign of its Gauss curvature, and its global topological structure. At the end of the paper are made some brief remarks concerning the possibility of applying analogous methods to the study of n-dimensional manifolds.
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Bibliografia
B. v. Kerékjártó,Vorlesungen über Topologie (Berlin 1923).
H. Hopf undW. Rinow,Über den Begriff der vollständigen differentialgeome-trischen Fläche, Comment. Math. Helvet. 3 (1931), pp. 209–225.
W. Blaschke,Vorlesungen über Differentialgeometrie I (3. Auflage, Berlin 1930).
S. Cohn-Vossen,Vollständige Riemannsche Räume positiver Krümmung, C. R. Acad. Sci. U.R.S.S. 3 (1935), pp. 387–389.
A. Preissmann,Quelques propriétés globales des espaces de Riemann, Comment. Math. Helvet. 15 (1943), pp. 175–216.
W. Rinow,Über Zusammenhänge zwischen der Differentialgeometrie in Grossen und im Kleinen, Math. Zeitschrift 35 (1932), pp. 512–538.
E. Cartan,Leçons sur la géométrie des espaces de Riemann (2. édition, Paris 1946).
S. Cohn-Vossen,Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), pp. 69–133.
S. Cohn-Vossen,Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken, Recueil Math. (nouvelle série) 1 (1936), pp. 139–163.
S. B. Myers,Riemannian manifolds in the large, Duke Math. Journ. 1 (1935), pp. 39–49.
C. B. Allendoerfer andAndré Weil,The Gauss-Bonnet theorem for Riemannian polyhedra, Transact. Amer. Math. Soc. 53 (1943), pp. 101–129.
S. S. Chern,A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Annals of Math. 45 (1944), pp. 747–752.
J. M. Schoenberg,Some applications of the calculus of variation to Riemannian geometry, Annals of Math. 33 (1932), pp. 485–495.
J. L. Synge,On the connectivity of spaces of positive curvature, Quarterly Journ. of Math. (Oxford series) 7 (1936), pp. 316–320.
S. Bochner,Vector fields and Ricci curvature, Bull. Amer. Math. Soc., 52 (1946), pp. 776–797. -Curvature in hermitian metric, Bull. Amer. Math. Soc. 53 (1946), pp. 179–195.
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Hopf, H. Sulla geometria riemanniana globale della superficie. Seminario Mat. e Fis. di Milano 23, 48–63 (1953). https://doi.org/10.1007/BF02922523
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DOI: https://doi.org/10.1007/BF02922523