Submanifolds In Space Forms Research Articles

A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications

Integral section formulae for totally geodesic submanifolds (planes) intersecting a compact submanifold in a space form are available from appropriate representations of the motion invariant density (measure) of these planes. Here we present a new decomposition of the invariant density of planes in space forms. We apply the new decomposition to rewrite Santaló's sectioning formula and thereby to obtain new mean values for lines meeting a convex body. In particular we extend to space forms a recently published stereological formula valid for isotropic plane sections through a fixed point of a convex body in R 3 .

Homology vanishing theorems for submanifolds

We relate intrinsic and extrinsic curvature invariants to the homology groups of submanifolds in space forms of nonnegative curvature. More precisely, we provide bounds for the squared length of the second fundamental form, or the Ricci curvature in terms of the mean curvature, which force homology to vanish in a range of intermediate dimensions. Moreover, we give examples which show that these conditions are sharp.

R-Minimal submanifolds in space forms

Let x: M → R n+p (c) be an n-dimensional compact, possibly with bound- ary, submanifold in an (n + p)-dimensional space form R n+p (c). Assume that r is even and r ∈{ 0,1, ... ,n − 1}, in this paper we introduce rth mean curvature function Sr and (r + 1)-th mean curvature vector field Sr+1 .W e callM to be an r-minimal submanifold if Sr+1 ≡ 0o nM, we note that the concept of 0-minimal submani- fold is the concept of minimal submanifold. In this paper, we define a functional Jr(x) = � M Fr(S0,S2, ... ,Sr)dv of x: M → R n+p (c), by calculation of the first varia- tional formula of Jr we show that x is a critical point of Jr if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We cal- culate the second variational formula of Jr and prove that there exists no compact without boundary stable r-minimal submanifold with Sr > 0 in the unit sphere S n+p . When r = 0, noting S0 = 1, our result reduces to Simons' result: there exists no compact without boundary stable minimal submanifold in the unit sphere S n+p .

Doubly ruled submanifolds in space forms

In this paper we extend a classical result, namely, the one that states that the only doubly ruled surfaces in $\mathbb R^3$ are the hyperbolic paraboloid and the hyperboloid of one sheet, in three directions: for all space forms, for any dimensions of the rulings and manifold, and to the conformal realm. We show that all this can be reduced, with the help of quite natural constructions, to just one simple example, the rank one real matrices. We also give the affine classification in Euclidean~space. To deal with the conformal case, we make use of recent developments on Ribaucour transformations.

The Ribaucour transformation in Lie sphere geometry

We discuss the Ribaucour transformation of Legendre (contact) maps in its natural context: Lie sphere geometry. We give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer–Tojeiro as well as a higher dimensional version with the combinatorics of a cube. We also show how these theorems descend to the corresponding results for submanifolds in space forms.

A splitting theorem for Kähler submanifolds of space-forms

Isometric immersions from Kahler manifolds with parallel pluri-mean curvature (ppmc) generalize, in a natural way, the constant mean curvature (cmc) sufaces. The (2,0) part of the complexified second fundamental form is a holomorphic quadratic differential (Q) which plays a central role in the study of the cmc sufaces. Likewise, for ppmc immersions, Q is also a vector bundle valued holomorphic quadratic differencial, significant in the study of the geometry of the immersion. It is well known that those immersions with Q vanishing are extrinsically symmetric ([10] and [11]). In this work we study ppmc immersions with big nullity index of Q.

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.

Codimension two Kähler submanifolds of space forms

Codimension two Kähler submanifolds of space forms

On Ribaucour transformations and applications to linear Weingarten surfaces

We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.

Cut loci of submanifolds in space forms and in the geometries of Möbius and lie

The cut locus of a submanifold embedded properly in Euclidean space, hyperbolic space, or the sphere has an elementary description in terms of the set of maximal supporting balls of the submanifold. Three applications are given. The first is a proof that the topology of the cut locus of a submanifold is invariant when the submanifold is subjected to a Mobius transformation. The second is a simple method for constructing Riemannian manifolds which have a point whose cut locus is nontriangulable. The third is an investigation of the behavior of the cut locus when a submanifold of a sphere is subjected to a transformation of Lie Sphere Geometry.

Submanifold geometry in symmetric spaces

The classical local invariants of a submanifold in a space form are the first fundamental form, the shape operators and the induced normal connection, and they determine the submanifold up to ambient isometry. One of the main topics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds. Many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions. The study of focal points of a submanifold in an arbitrary Riemannian manifold arises from the Morse theory of the energy functional on the space of paths in the Riemannian manifold joining a fixed point to the submanifold. The Morse index theorem relates the geometry of a submanifold to the topology of this path space. The focal structure is intimately related to the local invariants of the submanifold. In the case of space forms one can go backwards and reconstruct the local invariants from the focal structure, so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure. What is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade: while extending the theory of submanifolds to ambient spaces more general than space-forms proves quite difficult if one tries to use the same approach as for the space forms, at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that reduces to the classical theory in the case of space forms. This paper is an extended report on this theory, and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces. First we set up some notations. Let (N, g) be a Riemannian manifold, M an immersed submanifold of N , and ν(M) the normal bundle of M . The end point map η : ν(M) → N of M is the restriction of the exponential map exp to ν(M). If v ∈ ν(M)x is a singular point of η and the dimension of the kernel of dηv is m, then v is called a multiplicity m focal normal and exp(v) is called a multiplicity m focal point of M with respect to M in N . The focal data, Γ(M), is defined to be the set of all pairs (v, m) such that v is a multiplicity m focal normal of M . The focal variety V(M) is the set of all pairs (η(v), m) with (v, m) ∈ Γ(M). The main purpose of our paper is to study the global geometry and topology

Rigidity of minimal submanifolds in space forms

(1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal immersion g: M" ~ A4" + q(c), q >= p, then there exists a rigid motion T of M~ + q(c) such that g = To f, (~l"+P(c) being considered as a totally geodesic submanifold of

Some remarks on a class of submanifolds in space forms of non-negative curvature

Some remarks on a class of submanifolds in space forms of non-negative curvature