Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds , smooth manifolds with a Riemannian metric , i. This gives, in particular, local notions of angle , length of curves , surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture " Ueber die Hypothesen, welche der Geometrie zu Grunde liegen " "On the Hypotheses on which Geometry is Based". It is a very broad and abstract generalization of the differential geometry of surfaces in R 3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.
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We'd like to understand how you use our websites in order to improve them. Register your interest. 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.
This is a preview of subscription content, log in to check access. Hopf und W. Auflage, Berlin Google Scholar. Zeitschrift 35 , pp. Myers , Riemannian manifolds in the large , Duke Math. Schoenberg , Some applications of the calculus of variation to Riemannian geometry , Annals of Math. Synge , On the connectivity of spaces of positive curvature , Quarterly Journ.
Oxford series 7 , pp. Bochner , Vector fields and Ricci curvature , Bull. Download references. Reprints and Permissions. Hopf, H. Sulla geometria riemanniana globale della superficie. Seminario Mat. Download citation. Issue Date : December Search SpringerLink Search. Summary This lecture is a report upon a chapter of differential geometry in the large.
Bibliografia  B. Authors Heinz Hopf View author publications. You can also search for this author in PubMed Google Scholar. Rights and permissions Reprints and Permissions. About this article Cite this article Hopf, H.
Sulla geometria riemanniana globale della superficie