共查询到20条相似文献,搜索用时 15 毫秒
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A. D. Milka 《Journal of Mathematical Sciences》1986,33(2):910-933
We give a survey of results of the author on the geometry of geodesics and shortest curves on general convex hypersurfaces in spaces with constant curvature. We present qualitatively new theorems, which unite and generalize the main classical results; we give applications to the solution of a number of actual problems of geometry of convex surfaces in the large. We give generalizations of some well-known theorems on geodesics and shortest curves to Riemannian manifolds and nonregular multidimensional convex metrics.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 16, pp. 155–194, 1984. 相似文献
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Gerhard Huisken 《Inventiones Mathematicae》1986,84(3):463-480
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Let be two convex compact subsets of the hyperbolic space with smooth boundary. It is shown that the total curvature of the hypersurface is larger than the total curvature of .
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Given a null hypersurface L of a Lorentzian manifold, we construct a Riemannian metric on it from a fixed transverse vector field ζ. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold and the vector field ζ. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques. 相似文献
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Motivated by a conjecture of Steinhaus, we consider the mapping F, associating to each point x of a convex hypersurface the set of all points at maximal intrinsic distance from x. We first provide two large classes of hypersurfaces with the mapping F single-valued and involutive. Afterwards we show that a convex body is smooth and has constant width if its double has the above properties of F, and we prove a partial converse to this result. Additional conditions are given, to characterize the (doubly covered) balls. 相似文献
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Hanno Rund 《Annali di Matematica Pura ed Applicata》1971,88(1):99-122
The coefficients of the complete set of n fundamental forms of a hypersuface Vn−1 imbedded in an n-dimensional Riemannian space Vn, as recently introduced[(5)], are used to construct certain tensor fields over Vn−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply
in terms of polynomials involving the curvature tensor of Vn, the coefficients of the n fundamental forms, and the rth curvatures of Vn−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae
on Vn−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of
Vn−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above
vanishes identically whenever Vn is a space of constant curvature.
This research was supported by the National Research Council of Canada.
Entrata in Redazione il 21 agosto 1970. 相似文献
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Let be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of , which are those whose principal curvatures and angle function satisfy a relation , being W a differentiable function which is symmetric with respect to . When on the positive cone of , a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either or is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of and which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of and . 相似文献
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Jean-Pierre Ezin Marco Rigoli Isabel M. C. Salavessa 《Israel Journal of Mathematics》1990,71(2):193-209
We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR
n
to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally,
always exist and allow us to find interesting integral formulas fitting our purposes. 相似文献
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《European Journal of Operational Research》2020,280(3):1160-1170
We investigate the single-source-single-destination “shortest” path problem in directed, acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated path vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated path vectors from source to sink. 相似文献
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Ben Andrews James McCoy Yu Zheng 《Calculus of Variations and Partial Differential Equations》2013,47(3-4):611-665
We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface may persist, including flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become smooth and uniformly convex and contract to points. 相似文献
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Frank Morgan 《Transactions of the American Mathematical Society》2003,355(12):5041-5052
We add to the literature the well-known fact that an isoperimetric hypersurface of dimension at most six in a smooth Riemannian manifold is a smooth submanifold. If the metric is merely Lipschitz, then is still Hölder differentiable.
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V. A. Toponogov 《Siberian Mathematical Journal》1994,35(4):815-817
The research was supported by the Russian Foundation for Fundamental Research (Grant 93-011-179). 相似文献
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An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n+1)-space is Euclidean complete for n≥2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface
in R
3 must be an elliptic paraboloid.
Oblatum 16-VI-2001 & 27-II-2002?Published online: 29 April 2002 相似文献
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