共查询到20条相似文献,搜索用时 843 毫秒
1.
Barbara Opozda 《Monatshefte für Mathematik》2009,156(4):357-370
One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant
curvature in C
2 must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection.
In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds.
In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C
2. Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian)
geometry.
The research supported by the KBN grant 1 PO3A 034 26. 相似文献
2.
Barbara Opozda 《Geometriae Dedicata》2006,121(1):155-166
In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cn.
相似文献
3.
Barbara Opozda 《Geometriae Dedicata》1995,55(2):141-163
We present a new approach — which is more general than the previous ones — to the affine differential geometry of complex hypersurfaces inC
n+1. Using this general approach we study some curvature conditions for induced connections.The research supported by Alexander von Humboldt Stiftung and KBN grant no. 2 P30103004. 相似文献
4.
Fabio Podestà 《Results in Mathematics》1991,19(1-2):147-156
The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C2. 相似文献
5.
Thomas Binder 《Journal of Geometry》2004,79(1-2):31-45
We study non-degenerate affine surfaces in A3 with a projectively
flat induced connection. The curvature of the affine metric
, the affine
mean curvature H, and the Pick invariant
J are related by
.
Depending on the rank of the span of the gradients of these functions,
a local classification of three groups is given.
The main result is the characterization of the projectively flat but
not locally symmetric surfaces as a solution of a system of ODEs.
In the final part, we classify projectively flat and
locally symmetric affine translation surfaces. 相似文献
6.
Bang‐Yen Chen 《Mathematische Nachrichten》2005,278(11):1242-1281
A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of AH . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP 2 as well as in complex hyperbolic plane CH 2. We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP 2 and 41 families in CH 2. All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP 2 or in CH 2 is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
Friedrich Manhart 《Journal of Geometry》2004,80(1-2):166-178
We give a classification of affine rotational surfaces in affine 3-space with vanishing affine
Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves. 相似文献
8.
Hans Sachs 《Journal of Geometry》1976,8(1-2):33-47
In this paper we consider cyclic surfaces in an isotropic space of degree two, i.e. a three dimensional real affine space with the metric ds2=dx2. The local differential geometry of the first and second order of there surfaces is developed. An invariant moving frame is constructed by means of which some interesting questions of the extensive theory are solved.
Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag 相似文献
Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag 相似文献
9.
Rolf Walter 《Results in Mathematics》1995,27(1-2):194-197
For submanifolds of the affine space Rn, it is very important to derive a riemannian or pseudo-riemannian metric on the manifold just from affine data of the configuration. It is in this way that the equi-affine hypersurface theory is initiated by the so called Blaschke-Berwald metric (for the most recent state of affine hypersurface theory see the book of Li-Simon-Zhao [1993] and the vast literature given there). The same is true for the centro-affine geometry of codimension-two submanifolds (cf. Walter [1988], [1991 a]). Another instance where such a metric has been constructed from affine data are the (two-dimensional) surfaces of R4 (Burstin-Mayer [1927]). Recently, the geometry of these surfaces has been taken up by Nomizu-Vrancken [1993] with respect to the construction of a new transversal plane bundle. In the present note, we deal with the existence and, in particular, non-existence of elliptic points of the Burstin-Mayer metric from a local and global viewpoint. 相似文献
10.
Helmut Pabel 《Journal of Geometry》1991,40(1-2):148-164
We discuss with equiaffine methods the surfaces of translation with plane generating curves in the three-dimensional affine space. Using (pseudo-) isothermic parameters we determine in this class all the affine minimal surfaces (which include the affine spheres, the quadrics and the ruled surfaces), all the surfaces with vanishing affine Gauss curvature (which include the surfaces with constant non vanishing affine mean curvature), and all the surfaces with only one family of affine lines of curvature. 相似文献
11.
Daniele Sepe 《Geometriae Dedicata》2010,149(1):347-362
This paper completes the classification of regular Lagrangian fibrations over compact surfaces. Mishachev (Diff Geom Appl
6:301–320, 1996) classifies regular Lagrangian fibrations over
\mathbbT2{\mathbb{T}^2}. The main theorem in Fried et al. (Comment Math Helv 56(4):487–523, 1981) is used to in order to classify integral affine
structures on the Klein bottle K
2 and, hence, regular Lagrangian fibrations over this space. 相似文献
12.
Luc Vrancken 《Geometriae Dedicata》1990,33(2):177-194
In this paper, we study affine non-degenerate Blaschke immersions from a surface M in 3. We will assume that M has constant affine curvature and constant affine mean curvature, i.e. both the determinant and the trace of the shape operator are constant. Clearly, affine spheres satisfy both these conditions. In this paper, we completely classify the affine surfaces with constant affine curvature and constant affine mean curvature, which are not affine spheres.Research Assistant of the National Fund for Scientific Research (Belgium). 相似文献
13.
Francisco J. López 《Journal of Geometric Analysis》2014,24(2):988-1006
We prove a general fusion theorem for complete orientable minimal surfaces in ?3 with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries. 相似文献
14.
Ross Niebergall Patrick J. Ryan 《Transactions of the American Mathematical Society》1996,348(3):1093-1115
In this paper we study nondegenerate affine surfaces in whose affine principal curvatures are constant along their lines of curvature. We give a complete local classification of these surfaces assuming that the lines of curvature are planar, and there are no umbilics.
15.
A. Lesfari 《Results in Mathematics》1999,35(1-2):107-118
In C6, we consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points. 相似文献
16.
Abstract—In this paper, we consider connected minimal surfaces in R3 with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k g 1 and k g 2 of the coordinates curves satisfy αk g 1 + βk g 2 = 0, α, β ∈ R. 相似文献
17.
In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel. 相似文献
18.
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems.
We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems
which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic
topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric
proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.
相似文献
19.
Tong Zhu LI 《数学学报(英文版)》2005,21(6):1525-1534
Let f : M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L(f) = f(H2 - K)/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form. 相似文献
20.
Prof. Dr. Hans Sachs 《Monatshefte für Mathematik》1981,91(4):299-311
In this paper we consider complexes of lines in an isotropic space of degree two, i. e. a three dimensional real affine space with the metricds
2=dx
2. Using the method of differential forms we study the local differential geometry of first order and the theory of complex curves. Finally we give some applications in the theory of linear complexes. 相似文献