Euclidean Submanifold Research Articles

Some classification of affine homothetical surfaces of finite type in 𝕀

Abstract A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian Δ. In this paper, we classify two types of affine homothetical surfaces of finite type in isotropic 3-space 𝕀 3 {\mathbb{I}^{3}} under the condition Δ ⁢ r i = λ i ⁢ r i {\Delta r_{i}=\lambda_{i}r_{i}} , where ( r i ) {(r_{i})} is the i-component function of the position vector r ( i = 1 , 2 , 3 {i=1,2,3} ), λ i ∈ ℝ {\lambda_{i}\in\mathbb{R}} and Δ denotes the Laplace operator.

Helicoidal surfaces of finite type in the 3-dimensional Heisenberg group

An Euclidean submanifold is said to be of finite Chen-type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian [6]. In this article, we study helicoidal invariant surfaces in the 3-dimensional Heisenberg group H 3, which has finite type immersion.

Complete submanifolds with relative nullity in space forms

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor’s conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.

Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons.

Differential Geometry of Rectifying Submanifolds

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the notion of rectifying submanifolds in Euclidean spaces. In particular, we prove that a Euclidean submanifold is rectifying if and only if the tangential component of its position vector field is a concurrent vector field. Moreover, rectifying submanifolds with arbitrary codimension are completely determined.

Topics in differential geometry associated with position vector fields on Euclidean submanifolds

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The purpose of this article is to survey six research topics in differential geometry in which the position vector field plays very important roles. In this article we also explain the relationship between position vector fields and mechanics, dynamics, and D’Arcy Thompson’s law of natural growth in biology.

Cylindricity of complete Euclidean submanifolds with relative nullity

We show that a complete Euclidean submanifold with minimal index of relative nullity \(\nu _0>0\) and Ricci curvature with a certain controlled decay must be a \(\nu _0\)-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Normal holonomy of orbits and Veronese submanifolds

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let $M^n$, $n\geq2$, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset \mathbb{R}^N$ such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then $M^n$ must be an orbit of an irreducible $s$-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If $n=2$, then the normal holonomy is always transitive, unless $M$ is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when $n=3$. In this case $M^3$ must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for $n\geq 3$ when the normal holonomy acts irreducibly and the codimension is the maximal possible $n(n+1)/2$. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case $n=3$, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold $M^n$, is less or equal than $[n/2]$ (which is the rank of the orthogonal group $SO(n)$). This bound is sharp and improves the known bound $n(n-1)/2$.

The associated family of an elliptic surface and an application to minimal submanifolds

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a certain order are circles. We also provide the conditions under which this associated family is trivial, extending the known result for minimal surfaces. As an application, we show how the associated family of a minimal Euclidean submanifold of rank two is determined by the associated family of an elliptic surface clarifying the geometry around the associated family of these higher dimensional submanifolds.

The Contact Number of an Affine Immersion and Its Upper Bounds

We generalize the contact number of a Euclidean submanifold to that of an affine immersion. We show that when the ambient space is projectively flat if the contact number of an affine immersion with codimension p is finite, then it is at most p + 1.

Some upper bounds for sums of eigenvalues of the Neumann Laplacian

Let μ k (Ω) be the kth Neumann eigenvalue of a bounded domain Ω with piecewisely smooth boundary in R. In 1992, P. Kroger proved that k - n+2 n Σ k j=1 μ j ≤ 4nπ 2 Ν+2(ω n V) -2/n , where the upper bound is sharp in view of Weyl's asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of k to its right-hand side which approaches strictly from below to 1 as k tends to infinity. Second, we will generalize Kroger's estimate to the case when Ω is a compact Euclidean submanifold.

Surfaces with high contact number and their characterization

The notion of contact number $${\mathfrak c}_\#(M)$$ of a Euclidean submanifold was introduced in an earlier article (Proc. Edinb. Math. Soc. 47:69–100, 2004) as the highest order of contact of geodesics and normal sections on the submanifold. It was proved in (Proc. Edinb. Math. Soc. 47:69–100, 2004) that the contact number relates closely with the notions of isotropic submanifolds and holomorphic curves. One important problem concerning contact number is to construct Euclidean submanifolds with high contact number. The purpose of this article is thus to construct Euclidean surfaces with high contact number and to provide simple geometric characterization of such surfaces.

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.

Spacetime structure of static solutions in Gauss-Bonnet gravity: Neutral case

We study the spacetime structures of the static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-$\ensuremath{\Lambda}$ system systematically. We assume the Gauss-Bonnet coefficient $\ensuremath{\alpha}$ is non-negative and a cosmological constant is either positive, zero, or negative. The solutions have the $(n\ensuremath{-}2)$-dimensional Euclidean submanifold, which is the Einstein manifold with the curvature $k=1$, 0, and $\ensuremath{-}1$. We also assume $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}\ensuremath{\le}1$, where $\ensuremath{\ell}$ is the curvature radius, in order for the sourceless solution ($M=0$) to be defined. The general solutions are classified into plus and minus branches. The structures of the center, horizons, infinity, and the singular point depend on the parameters $\ensuremath{\alpha}$, ${\ensuremath{\ell}}^{2}$, $k$, $M$, and branches complicatedly so that a variety of global structures for the solutions are found. In our analysis, the $\stackrel{\texttildelow{}}{M}\mathrm{\text{\ensuremath{-}}}r$ diagram is used, which makes our consideration clear and enables easy understanding by visual effects. In the plus branch, all the solutions have the same asymptotic structure at infinity as that in general relativity with a negative cosmological constant. For the negative-mass parameter, a new type of singularity called the branch singularity appears at nonzero finite radius $r={r}_{b}>0$. The divergent behavior around the singularity in Gauss-Bonnet gravity is milder than that around the central singularity in general relativity. There are three types of horizons: inner, black hole, and cosmological. In the $k=1,0$ cases, the plus-branch solutions do not have any horizon. In the $k=\ensuremath{-}1$ case, the radius of the horizon is restricted as ${r}_{h}<\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$ (${r}_{h}>\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$) in the plus (minus) branch. The black hole solution with zero or negative mass exists in the plus branch even for the zero or positive cosmological constant. There is also the extreme black hole solution with positive mass. We briefly discuss the effect of the Gauss-Bonnet corrections on black hole formation in a collider and the possibility of the violation of the third law of the black hole thermodynamics.

On real Kähler Euclidean submanifolds with non-negative Ricci curvature

We show that any real Kähler Euclidean submanifold f:M^{2n} \to \mathbb R^{2n+p} with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to 2n-2p . Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that M^{2n} is complete. In particular, we conclude that the only real Kähler submanifolds M^{2n} in \R^{3n} that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of n orientable surfaces in \R^3 with positive Gaussian curvature. Further applications of our main result are also given.

A local and global splitting result for real K�hler Euclidean submanifolds

We show that if a real Kahler Euclidean submanifold is as far as possible of being minimal, then it should split locally as a product of hypersurfaces almost everywhere, possibly in lower codimension. In addition, if the manifold is complete, simply connected and has constant nullity, it should split globally as a product of surfaces in % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFDeIudaahaaWcbeqa % aiaaiodaaaaaaa!4190! $$\mathbb{R}^3 $$ and an Euclidean factor. Several applications are also given.

Genuine Rigidity of Euclidean Submanifolds in Codimension Two

An isometric deformation of an Euclidean submanifold is called genuine if the submanifold cannot be included into a submanifold of larger dimension in such a way that the deformation of the former is given by an isometric deformation of the latter. The submanifold is said to be genuinely rigid if it has no genuine deformations. In this paper we study the deformation problem in codimension two for the classes of elliptic and parabolic submanifolds. In spite of having a second fundamental form as degenerate as possible without being flat, i.e., the Gauss map has rank two everywhere, our main result says that generically these submanifolds are genuinely rigid. An additional unexpected deformation phenomenon for elliptic submanifolds carrying a Kaehler structure is described.

THE CONTACT NUMBER OF A EUCLIDEAN SUBMANIFOLD

AbstractWe introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-$2$ submanifolds with contact number ${\geq}3$. We also study surfaces in $\mathbb{E}^6$ with contact number ${\geq}4$. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.AMS 2000 Mathematics subject classification: Primary 53C40; 53A10. Secondary 53B25; 53C42

Genuine Deformations of Submanifolds

We introduce the concept of genuine isometric deformation of an Euclidean submanifold and describe the geometric structure of the submanifolds that admit deformations of this kind. That an isometric deformation is genuine means that the submanifold is not included into a submanifold of larger dimension such that the deformation of the former is given by a deformation of the latter. Our main result says that an Euclidean submanifold together with a genuine deformation in low (but not necessarily equal) codimensions must be mutually ruled, and gives a sharp estimate for the dimension of the rulings. This has several strong local and global consequences. Moreover, the unifying character and geometric nature, as opposed to a purely algebraic one, of our result suggest that it should be the starting point for a deformation theory extending the classical Sbrana Cartan theory for hypersurfaces to higher codimensions. The isometric deformation problem for a given isometric immersion f : M → R of a Riemannian manifold into flat Euclidean space with codimension p and a positive integer q is to describe all possible isometric immersions f : M → R. A satisfactory answer to the local version of the problem for every hypersurface (p=1) and q = 1 going back almost a century is due to Sbrana [19] and Cartan [4]. However, basic questions, like the existence of Sbrana-Cartan hypersurfaces of the discrete type or the possibility of smoothly attaching different types of these deformable hypersurfaces, were answered positively only recently; see [10] and also [3] for a special case. The global version of the problem for hypersurfaces was solved in [11] and [18]. Nothing similar to the Sbrana-Cartan theory for codimensions q = p higher than one has yet been obtained. Nevertheless, the classical Beez-Killing rigidity theorem for hypersurfaces, the starting point for the theory, has several generalizations; see [1], [2], [5] and [20]. All these results provide generic algebraic conditions on the second fundamental form of the isometric immersion that imply isometric rigidity, that is, any other isometric immersion must differ by an isometry (rigid motion) of the ambient space. There is a large set of isometric deformations once we allow codimension q > p since one can always compose f : M → R with (local) isometric immersions of R into R. In [7] we found generic algebraic conditions on the second fundamental form

A class of austere submanifolds

Austerity is a pointwise algebraic condition on the second fundamental form of an Euclidean submanifold and requires that the nonzero principal curvatures in any normal direction occur in pairs with opposite signs. These submanifolds have been introduced by Harvey and Lawson in the context of special Lagrangian submanifolds. The main purpose of this paper is to classify all austere submanifolds whose Gauss maps have rank two. This condition means that the image of the Gauss map in the corresponding Grassmannian is a surface. The hypersurface case is due to Dajczer and Gromoll and the three dimensional case to Bryant. We show that any such submanifold is, roughly, a subbundle of the normal bundle of a surface whose ellipse of curvature of a certain order is a circle. We also characterize austere submanifolds which are Kaehler manifolds.