In this paper, we study hypersurfaces of the homogeneous nearly Kähler manifold S 3 ×S 3 with typical properties. We first show that in the NK S 3 ×S 3 there exist neither totally umbilical ...hypersurfaces nor hypersurfaces of parallel second fundamental form. Then we investigate hypersurfaces of S 3 ×S 3 such that its shape operator A and induced almost contact structure satisfy the condition A = A, and as the main result, a complete classification of this remarkable family of hypersurfaces in S 3 ×S 3 is presented.
In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. ...Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.
The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation ...between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.
This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that ...combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic.
We prove a theorem of Hadamard–Stoker type: a connected locally convex complete hypersurface immersed in H n × R ( n ≥ 2 ), where H n is n-dimensional hyperbolic space, is embedded and homeomorphic ...either to the n-sphere or to R n . In the latter case it is either a vertical graph over a convex domain in H n or has what we call a simple end.
In this paper, we prove there exist at least n+12+1 geometrically distinct closed characteristics on every compact convex hypersurface Σ in R2n, where n≥2. In particular, this gives a new proof in ...the case n=3 to a long standing conjecture in Hamiltonian analysis. Moreover, there exist at least n2+1 geometrically distinct non-hyperbolic closed characteristics on Σ provided the number of geometrically distinct closed characteristics on Σ is finite.
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