Theory Of Submanifolds Research Articles

Locally symmetric minimal affine Lagrangian surfaces in C2

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C2 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 C2. Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry.

Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold M endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure θ on M , [20]) on the total space of the canonical circle bundle S 1 →C(M) → π M (a manifold with boundary ∂C(M)= π −1 (∂M) and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N={φ=0}⊂ H 1 we show that the mean curvature vector of N⇀ H 1 is expressed by H=− 1 2 ∑ j=1 2 X j (|Xφ | −1 X j φ)ξ provided that N is tangent to the characteristic direction T of ( H 1 , θ 0 ) , thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion Ψ:N→ H n of a Riemannian manifold into the Heisenberg group we show that ΔΨ=2J T ⊥ hence start a Weierstrass representation theory for minimal surfaces in H n .

Characteristic subsurfaces, character varieties and Dehn fillings

We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.

RICCI CURVATURE OF INTEGRAL SUBMANIFOLDS OF AN S-SPACE FORM

Abstract. Involving the Ricci curvature and the squared mean curva-ture, we obtain a basic inequality for an integral submanifold of an S -space form. By polarization, we get a basic inequality for Ricci tensoralso. Equality cases are also discussed. By giving a very simple proof weshow that if an integral submanifold of maximum dimension of an S -spaceform satisfles the equality case, then it must be minimal. These resultsare applied to get corresponding results for C -totally real submanifoldsof a Sasakian space form and for totally real submanifolds of a complexspace form. 1. IntroductionOne of the most fundamental problems in submanifold theory is the follow-ing: Establish simple relationships between the main extrinsic invariants andthe main intrinsic invariants of a submanifold. In [7], B.-Y. Chen established asharp relationship between the Ricci curvature and the squared mean curvaturefor a submanifold in a Riemannian space form with arbitrary codimension. In[8], he gave the corresponding version of this inequality for totally real sub-manifolds in a complex space form. We flnd corresponding results for

Theory of submanifolds, associativity equations in 2D topological quantum field theories, and Frobenius manifolds

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove in this paper that each N-dimensional Frobenius manifold can locally be represented as a potential flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction this submanifold is uniquely determined up to motions. Moreover, in this paper we consider a nonlinear system, which is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method.

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.

Chen's inequality in the Lagrangian case

In the theory of submanifolds, the following problem is fundamental: es- tablish simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds. The basic relationships discovered until now are inequalities. To analyze such problems, we follow the idea of C. Udriste that the method of constrained extremum is a natural way to prove geometric inequalities. We improve Chen's inequal- ity which characterizes a totally real submanifold of a complex space form. For that we suppose that the submanifold is Lagrangian and we formulate and analyze a suitable constrained extremum problem.

Deformation of special Lagrangian suborbifolds

The purpose of this paper is to generalize the deformation theory of special Lagrangian submanifolds developed by Mclean and Hitchin to special Lagrangian suborbifolds.

A pinching theorem for the normal scalar curvature of invariant submanifolds

We prove some inequalities relating intrinsic and extrinsic curvature invariants for invariant submanifolds of Kaehlerian and Sasakian space forms. When we restrict to invariant submanifolds of odd-dimensional unit spheres or invariant submanifolds of complex Euclidean space, one of the inequalities gives a positive answer to a conjecture, proposed in [P.J. De Smet, F. Dillen, L. Verstraelen, L. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. (Brno) 35 (1999) 115–128].

Deformation and applicability of surfaces in Lie sphere geometry

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations, and the corresponding existence problem is discussed.

Construction of Lagrangian surfaces in complex Euclidean plane with Legendre curves

An important problem in the theory of Lagrangian submanifolds is to find non-trivial examples of Lagrangian submanifolds in complex Euclidean spaces with some given special geometric properties. In this article, we provide a new method to construct Lagrangian surfaces in the complex Euclidean plane C2 by using Legendre curves in S3(1) $\subset$ C2. We also investigate intrinsic and extrinsic geometric properties of the Lagrangian surfaces in C2 obtained by applying our construction method. As an application we provide some new families of Hamiltonian minimal Lagrangian surfaces in C2 via our construction method.

Isothermic submanifolds of Euclidean space

We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold isothermic if it is locally the image of a conformal immersion of a Riemannian product of Riemannian manifolds whose second fundamental form is adapted to the product net of the manifold. Our main result is a complete classification of all such conformal immersions of Riemannian products of dimension greater than or equal to three. We derive several consequences of this result. We also study whether the classical characterizations of isothermic surfaces as solutions of Christoffel’s problem and as envelopes of nontrivial conformal sphere congruences extend to higher dimensions. MSC 2000: 53 C15, 53 C12, 53 C50, 53 B25.

Non-isotopic Legendrian submanifolds in R2n+1

In the standard contact (2n + 1)-space when n > 1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using classically known invariants. When n is even, these are the first known examples of non-Legendrian isotopic, Legendrian submanifolds of (2n + 1)-space. Such constructions indicate a rich theory of Legendrian submanifolds. To distinguish our examples, we compute their contact homology which was rigorously defined in this situation in [7].

Characteristic subsurfaces and Dehn filling

Let M M be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in M M which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of M ( α ) M(\alpha ) has no non-abelian free subgroup, and if M ( β ) M(\beta ) is a reducible manifold which is not homeomorphic to S 1 × S 2 S^1 \times S^2 or P 3 # P 3 P^3 \# P^3 , then Δ ( α , β ) ≤ 5 \Delta (\alpha , \beta )\le 5 . Under the same condition on M ( β ) M(\beta ) , it is shown that if M ( α ) M(\alpha ) is Seifert fibered, then Δ ( α , β ) ≤ 6 \Delta (\alpha , \beta )\le 6 . Moreover, in the latter situation, character variety techniques are used to characterize the topological types of M ( α ) M(\alpha ) and M ( β ) M(\beta ) in case the bound of 6 6 is attained.

Compatible metrics on a manifold and nonlocal bi-Hamiltonian structures

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated non-local bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasi-homogeneity assumptions one arrives at a so-called weak $\F$-manifold - a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.

Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups

Introduction Johannson's characteristic submanifold theory Relative compression bodies and cores Homotopy types Pared 3-manifolds Small 3-manifolds Geometrically finite hyperbolic 3-manifolds Statements of main theorems The case when there is a compressible free side The case when the boundary pattern is useful Dehn flips Finite index realization for reducible 3-manifolds Epilogue Bibliography Index.

Complete, embedded, minimal n‐dimensional submanifolds in ℂn

A classical problem in the theory of minimal submanifolds of Euclidean spaces is to understand whether a minimal submanifold exists with a prescribed behavior at infinity, or to determine from the asymptotes the geometry of the whole submanifold. Beyond the intrinsic interest of these questions, they are also of crucial importance when studying the possible singularities of minimal submanifolds in general Riemannian manifolds. For minimal surfaces, i.e., two-dimensional submanifolds, the standard tool to solve these problems is given by the Weierstrass representation formula, which relates the geometry of the minimal surface to complex analytic properties of holomorphic 1-forms on Riemann surfaces. Recently gluing techniques have been developed that provide an abundant number of new examples of minimal hypersurfaces in Euclidean space. For higher-dimensional minimal submanifolds such a link clearly disappears and no complex analysis can be put into play. Although gluing techniques have been extensively used in the study of minimal hypersurfaces, they have not been adapted to handle higher codimensional submanifolds. The aim of this paper is to use a gluing technique for minimal submanifolds to make a step toward understanding these questions in higher codimension. More precisely, we will restrict ourselves to the case of real n-dimensional submanifolds of Cn . There are two main reasons for doing so. The first one comes from the fact that when trying to desingularize the intersection, for example, of a pair of n-planes that give the desired asymptotic behavior, one needs a model of a minimal submanifold with this behavior at infinity to rescale and to glue into the pair of planes where a neighborhood of the intersection is removed. This local model needs to be sufficiently simple to allow a detailed study of the linearized mean curvature operator. In our situation this is provided by a generalization of an area-minimizing submanifold found by Lawlor in [12], while in more general cases such an example is not known. The second reason is that, among minimal n-submanifolds ofCn , there is a

A totally geodesic property of Hopf vector fields

We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.

An Extension of The Classical Ribaucour Transformation

We extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with flat normal bundle admitting a global system of principal coordinates. Bianchi gave a positive answer to the question of whether among the Ribaucour transforms of a surface with constant mean or Gaussian curvature there exist other surfaces with the same property. Our main achievement is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature. 2000 Mathematical Subject Classification: 53B25, 58J72.

The Conjugate Nets, Cartan Submanifolds, and Laplace Transformations in Space Forms

In this paper we establish the geometric theory of conjugate nets, Cartan submanifolds, and Laplace transformations in sphere and pseudo-sphere spaces. The corresponding theory in cases of projective and Euclidean spaces has been established by Chern, Kamran and Tenenblat.