Mutual Position of Hypersurfaces in Projective Space
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Complete Hypersurfaces in Euclidean Spaces with Finite Strong Total
Complete hypersurfaces in Euclidean spaces with finite strong total curvature Manfredo do Carmo and Maria Fernanda Elbert Abstract We prove that finite strong total curvature (see definition in Section 2) complete hypersurfaces of (n + 1)-euclidean space are proper and diffeomorphic to a com- pact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punc- tures. This is related to results of White [22] and and M¨uller-Sver´ak[18].ˇ Further properties of these hypersurfaces are presented, including a gap theorem for the total curvature. 1 Introduction Let φ: M n ! Rn+1 be a hypersurface of the euclidean space Rn+1. We assume that n n n+1 M = M is orientable and we fix an orientation for M. Let g : M ! S1 ⊂ R be the n Gauss map in the given orientation, where S1 is the unit n-sphere. Recall that the linear operator A: TpM ! TpM, p 2 M, associated to the second fundamental form, is given by hA(X);Y i = −hrX N; Y i; X; Y 2 TpM; where r is the covariant derivative of the ambient space and N is the unit normal vector in the given orientation. The map A = −dg is self-adjoint and its eigenvalues are the principal curvatures k1; k2; : : : ; kn. Key words and sentences: Complete hypersurface, total curvature, topological properties, Gauss- Kronecker curvature 2000 Mathematics Subject Classification. 57R42, 53C42 Both authors are partially supported by CNPq and Faperj. 1 R n We say that the total curvature of the immersion is finite if M jAj dM < 1, where P 21=2 n n n+1 jAj = i ki , i.e., if jAj belongs to the space L (M). -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Linear Subspaces of Hypersurfaces
LINEAR SUBSPACES OF HYPERSURFACES ROYA BEHESHTI AND ERIC RIEDL Abstract. Let X be an arbitrary smooth hypersurface in CPn of degree d. We prove the de Jong-Debarre Conjecture for n ≥ 2d−4: the space of lines in X has dimension 2n−d−3. d+k−1 We also prove an analogous result for k-planes: if n ≥ 2 k + k, then the space of k- planes on X will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying n ≥ 2d! is unirational, and we prove that the space of degree e curves on X will be irreducible of the expected dimension provided that e+n d ≤ e+1 . 1. Introduction We work throughout over an algebraically closed field of characteristic 0. Let X ⊂ Pn be an arbitrary smooth hypersurface of degree d. Let Fk(X) ⊂ G(k; n) be the Hilbert scheme of k-planes contained in X. Question 1.1. What is the dimension of Fk(X)? In particular, we would like to know if there are triples (n; d; k) for which the answer depends only on (n; d; k) and not on the specific smooth hypersurface X. It is known d+k classically that Fk(X) is locally cut out by k equations. Therefore, one might expect the d+k answer to Question 1.1 to be that the dimension is (k + 1)(n − k) − k , where negative dimensions mean that Fk(X) is empty. This is indeed the case when the hypersurface X is general [10, 17]. Standard examples (see Proposition 3.1) show that dim Fk(X) must depend on the particular smooth hypersurface X for d large relative to n and k, but there remains hope that Question 1.1 might be answered positively for n large relative to d and k. -
Solutions to the Maximal Spacelike Hypersurface Equation in Generalized Robertson-Walker Spacetimes
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 80, pp. 1{14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO THE MAXIMAL SPACELIKE HYPERSURFACE EQUATION IN GENERALIZED ROBERTSON-WALKER SPACETIMES HENRIQUE F. DE LIMA, FABIO´ R. DOS SANTOS, JOGLI G. ARAUJO´ Communicated by Giovanni Molica Bisci Abstract. We apply some generalized maximum principles for establishing uniqueness and nonexistence results concerning maximal spacelike hypersur- faces immersed in a generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so-called timelike convergence condition (TCC). As application, we study the uniqueness and nonexistence of entire solutions of a suitable maximal spacelike hypersurface equation in GRW spacetimes obeying the TCC. 1. Introduction In the previous decades, the study of spacelike hypersurfaces immersed in a Lorentz manifold has been of substantial interest from both physical and mathe- matical points of view. For instance, it was pointed out by Marsden and Tipler [24] and Stumbles [37] that spacelike hypersurfaces with constant mean curvature in a spacetime play an important role in General Relativity, since they can be used as initial hypersurfaces where the constraint equations can be split into a linear system and a nonlinear elliptic equation. From a mathematical point of view, spacelike hypersurfaces are also interesting because of their Bernstein-type properties. One can truly say that the first remark- able results in this branch were the rigidity theorems of Calabi [13] and Cheng and Yau [15], who showed (the former for n ≤ 4, and the latter for general n) that the only maximal (that is, with zero mean curvature) complete spacelike hypersurfaces of the Lorentz-Minkowski space Ln+1 are the spacelike hyperplanes. -
The Projective Geometry of the Spacetime Yielded by Relativistic Positioning Systems and Relativistic Location Systems Jacques Rubin
The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques Rubin To cite this version: Jacques Rubin. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems. 2014. hal-00945515 HAL Id: hal-00945515 https://hal.inria.fr/hal-00945515 Submitted on 12 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques L. Rubin (email: [email protected]) Université de Nice–Sophia Antipolis, UFR Sciences Institut du Non-Linéaire de Nice, UMR7335 1361 route des Lucioles, F-06560 Valbonne, France (Dated: February 12, 2014) As well accepted now, current positioning systems such as GPS, Galileo, Beidou, etc. are not primary, relativistic systems. Nevertheless, genuine, relativistic and primary positioning systems have been proposed recently by Bahder, Coll et al. and Rovelli to remedy such prior defects. These new designs all have in common an equivariant conformal geometry featuring, as the most basic ingredient, the spacetime geometry. In a first step, we show how this conformal aspect can be the four-dimensional projective part of a larger five-dimensional geometry. -
Rotational Hypersurfaces Satisfying ∆ = in the Four-Dimensional
POLİTEKNİK DERGİSİ JOURNAL of POLYTECHNIC ISSN: 1302-0900 (PRINT), ISSN: 2147-9429 (ONLINE) URL: http://dergipark.org.tr/politeknik Rotational hypersurfaces satisfying ∆푰퐑 = 퐀퐑 in the four-dimensional Euclidean space Dört-boyutlu Öklid uzayında ∆퐼푹 = 푨푹 koşulunu sağlayan dönel hiperyüzeyler Yazar(lar) (Author(s)): Erhan GÜLER ORCID : 0000-0003-3264-6239 Bu makaleye şu şekilde atıfta bulunabilirsiniz (To cite to this article): Güler E., “Rotational hypersurfaces satisfying ∆퐼퐑 = 퐀퐑 in the four-dimensional Euclidean space”, Politeknik Dergisi, 24(2): 517-520, (2021). Erişim linki (To link to this article): http://dergipark.org.tr/politeknik/archive DOI: 10.2339/politeknik.670333 Rotational Hypersurfaces Satisfying 횫퐈퐑 = 퐀퐑 in the Four-Dimensional Euclidean Space Highlights ❖ Rotational hypersurface has zero mean curvature iff its Laplace-Beltrami operator vanishing ❖ Each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero ❖ Laplace-Beltrami operator of the rotational hypersurface depends on its mean curvature and the Gauss map Graphical Abstract Rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. The Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. In addition, it is indicated that each element of the 4×4 order matrix A, which satisfies the condition 훥퐼푅 = 퐴푅, is zero, that is, the rotational hypersurface R is minimal. Aim We consider the rotational hypersurfaces in 피4 to find its Laplace-Beltrami operator. Design & Methodology We indicate fundamental notions of 피4. Considering differential geometry formulas in 3-space, we transform them in 4-space. -
Convex Sets in Projective Space Compositio Mathematica, Tome 13 (1956-1958), P
COMPOSITIO MATHEMATICA J. DE GROOT H. DE VRIES Convex sets in projective space Compositio Mathematica, tome 13 (1956-1958), p. 113-118 <http://www.numdam.org/item?id=CM_1956-1958__13__113_0> © Foundation Compositio Mathematica, 1956-1958, tous droits réser- vés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Convex sets in projective space by J. de Groot and H. de Vries INTRODUCTION. We consider the following properties of sets in n-dimensional real projective space Pn(n > 1 ): a set is semiconvex, if any two points of the set can be joined by a (line)segment which is contained in the set; a set is convex (STEINITZ [1]), if it is semiconvex and does not meet a certain P n-l. The main object of this note is to characterize the convexity of a set by the following interior and simple property: a set is convex if and only if it is semiconvex and does not contain a whole (projective) line; in other words: a subset of Pn is convex if and only if any two points of the set can be joined uniquely by a segment contained in the set. In many cases we can prove more; see e.g. -
Rotation Hypersurfaces in Sn × R and Hn × R
Note di Matematica Note Mat. 29 (2009), n. 1, 41-54 ISSN 1123-2536, e-ISSN 1590-0932 DOI 10.1285/i15900932v29n1p41 Notehttp://siba-ese.unisalento.it, di Matematica 29, ©n. 2009 1, 2009, Università 41–54. del Salento __________________________________________________________________ Rotation hypersurfaces in Sn × R and Hn × R Franki Dillen i [email protected] Johan Fastenakels ii [email protected] Joeri Van der Veken iii Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B Box 2400, BE-3001 Leuven, Belgium. [email protected] Received: 14/11/2007; accepted: 07/04/2008. Abstract. We introduce the notion of rotation hypersurfaces of Sn × R and Hn × R and we prove a criterium for a hypersurface of one of these spaces to be a rotation hypersurface. Moreover, we classify minimal rotation hypersurfaces, flat rotation hypersurfaces and rotation hypersurfaces which are normally flat in the Euclidean resp. Lorentzian space containing Sn ×R resp. Hn × R. Keywords: rotation hypersurface, flat, minimal MSC 2000 classification: 53B25 1 Introduction In[3]theclassicalnotionofarotationsurfaceinE3 was extended to rota- tion hypersurfaces of real space forms of arbitrary dimension. Motivated by the recent study of hypersurfaces of the Riemannian products Sn × R and Hn × R, see for example [2], [4], [5] and [6], we will extend the notion of rotation hyper- surfaces to these spaces. Starting with a curve α on a totally geodesic cylinder S1 × R resp. H1 × R and a plane containing the axis of the cylinder, we will construct such a hypersurface and we will compute its principal curvatures. Moreover, we will prove a criterium for a hypersurface of Sn × R resp. -
Systems of Lines Amd Planes on the Quadric
Systems of lines and planes on the quadric hypersurface in spaces of four and five dimensions Item Type text; Thesis-Reproduction (electronic) Authors Fish, Anne, 1924- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 14:55:47 Link to Item http://hdl.handle.net/10150/319188 SYSTEMS OF LINES■AMD PLANES ON THE QUADRIC HYPERSURFACE SPACES OF FOUR AND- FIVE DIMENSIONS - by Anne Fish A TLes is submitted to the faculty of the Department of Mathematics in partial fullfilment of the requirements for the degree of Master of Science in the Graduate College University of Arizona 1948 Approved: Direeto^ of Thesis t Date & R A& TABLE OF CONTENTS I? O (TU. O "fc X OH a a © © a © o © © © © o © o © © ©ooo oool Chapter I Geometry of M-Dirnensiona 1 Projective Space © » 2 II Ruled Surfaces in Three-Dimensional Projective Space © © © © © © © © © © © © 10 III Systems of Lines on a Quadric Surface in Three-Dimensional Projective Space © © « © 13 IV Systems of Lines on a Quadric Hypersurface in Four-Dimensional Projective Space « © © © 23 V Systems of Planes on a Quadric Hypersurface in a Projective Space of Five Dimensions © © 34 Bibliography © © © © © © © © © © © © © © © © © © © s © © © 4V 194189 INTRODUCTION It has long been common knowledge -
Positive Geometries and Canonical Forms
Prepared for submission to JHEP Positive Geometries and Canonical Forms Nima Arkani-Hamed,a Yuntao Bai,b Thomas Lamc aSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA bDepartment of Physics, Princeton University, Princeton, NJ 08544, USA cDepartment of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA Abstract: Recent years have seen a surprising connection between the physics of scat- tering amplitudes and a class of mathematical objects{the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra{which have been loosely referred to as \positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic sin- gularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of \positive geometries" and their associated \canonical forms" as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones{via \triangulation" on the one hand, and \push-forward" maps between geometries on the other. We present numerous examples of positive geome- tries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest \simplex-like" geometries and the richer \polytope-like" ones. -
Arxiv:2008.11018V1 [Math.DG]
UNIQUENESS OF HYPERSURFACES OF CONSTANT HIGHER ORDER MEAN CURVATURE IN HYPERBOLIC SPACE BARBARA NELLI, JINGYONG ZHU Abstract. We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson [12] to the embedded hypersurfaces with constant higher order mean curva- ture. Then we prove two Bernstein type results for immersed hypersurfaces under different assumptions. Last, we show the rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures. 1. Introduction In 1927, S. N. Bernstein proved that the only entire minimal graphs in R3 are planes. The analogous problem in higher dimension is known as Bernstein problem. Namely, given u : Rn → R a minimal graph, is the graph of u a flat hyperplane? It turns out that the answer is yes for n ≤ 7 and that has been settled down by a series of very significative papers [4, 11, 16, 24]. On the contrary, for n ≥ 8, one has the famous counterexamples by E. Bombieri, E. De Giorgi and E. Giusti [6]. Afterwards, many generalizations of Bernstein theorem have arised. As an example, we mention [23] where R. Schoen, L. Simon and S. T. Yau studied a Bernstein type theorem for stable minimal hypersurfaces. Later on, Bernstein type theorems for constant mean curvature hypersurfaces in Euclidean and in hyperbolic space Hn+1 have been studied. Let us give a very simple example in the Euclidean space. Does it exist an entire graph M in Rn+1 with constant mean curvature H =6 0? It is well known that the answer is no and here is the proof. -
Properties of Hypersurfaces Which Are Characteristic for Spaces of Constant
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze OLDRICHˇ KOWALSKI Properties of hypersurfaces which are characteristic for spaces of constant curvature Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 26, no 1 (1972), p. 233-245 <http://www.numdam.org/item?id=ASNSP_1972_3_26_1_233_0> © Scuola Normale Superiore, Pisa, 1972, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ PROPERTIES OF HYPERSURFACES WHICH ARE CHARACTERISTIC FOR SPACES OF CONSTANT CURVATURE OLD0159ICH KOWALSKI, Praha If is well-known that the spaces of constant curvature are characteri- zed among all Riemannian spaces by the following property : for any (n -1)- dimensional linear elements En-l of a Riemann space N (dim N = n > 3) there is a totally geodesic hypersurface which is tangent to En-1. (Cf. [1]). The purpose of this Note is to present a number of theorems of the above type ; only the requirement that our hypersurfaces should be totally geodesic will be replaced be another geometrical or analytical postu- lates. Umbilical points of hypersurfaces and so called « normal Bianchi identity >> play the leading part here.