Homotopy connectedness theorems for submanifolds of Sasakian manifolds

The homotopy connectedness theorem for invariant immersions in Sasakian manifolds with nonnegative transversal q-bisectional curvature is proved. Some Barth-Lefschetz type theorems for minimal submanifolds and (k, ?)-saddle submanifolds in Sasakian manifolds with positive transversal q-Ricci curvature are proved by using the weak (?-)asymptotic index. As a corollary, the Frankel type theorem is proved.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants

In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed. From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and 127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π _n , there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem (in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1 manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle (by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 3–15, 2007.     

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Asymptotic Geometry and Conformal Types of Carnot-Carathéodory Spaces

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot—Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further applications are discussed, such as the relation between local and global invertibility properties of quasiconformal immersions (the global homeomorphism theorem). Submitted: November 1997, revised: November 1998.     

Rings of invariants for representations of quivers

In this Note we compute the generators of the ring of invariants for quiver factorization problems, generalizing results of Le Bruyn and Procesi. In particular, we find a necessary and sufficient combinatorial criterion for the projectivity of the associated invariant quotients. Further, we show that the non-projective quotients admit open immersions into projective varieties, which still arise from suitable quiver factorization problems. To cite this article: M. Halic, M.S. Stupariu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given： The affine immersions with pointwise planar normal sections （with respect to the equiaffine transversal bundle） are parallel. This result is verified for surfaces in R4 and for immersions with the maximal codimension.     

A combinatorial analog of a theorem of F.J. Dyson

Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem and the case $n=2$ was proposed by Tucker in 1945. Numerous generalizations and applications of the lemma have appeared since then. In 2006 Meunier proved the lemma in its full generality in his PhD thesis. There are generalizations and extensions of the Borsuk–Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's lemma, we hope that this will lead to generalizations and applications and ultimately a combinatorial analog of Yang's theorem of which both Borsuk–Ulam and Dyson are special cases.     

Yet another proof of Marstrand’s theorem

In a paper from 1954 Marstrand proved that if K ⊂ ℝ² is a Borel set with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem, extending the techniques developed in our previous paper [9].     

Extending immersions of curves to properly immersed surfaces

Proper generic immersions of compact one-dimensional manifolds in surfaces are studied. Suppose an immersion γ of a collection of circles is given with an even number of double points in a closed surface G. Then γ extends to various proper immersions of surfaces in three-manifolds that are bounded by G. Some of these extensions do not have triple points. The minimum of the genera of the triple point free surfaces is an invariant of the curve. An algorithm to compute this invariant is given.Necessary and suffecient conditions determine if a given collection δ of immersed arcs in a surface F maps to the double points set of a proper immersion. In case the conditions are satisfied, an immersion of F into a three-manifold that depends on δ is constructed explicitly. In the process, the possible triple points of immersed surfaces in three-manifolds are categorized.The techniques are applied to find examples of curves in surfaces that do not bound immersed disks in any three-manifold.     

Higher-order nondegenerate immersions of manifolds

In this paper we show how the Smale-Hirsch theory of immersions can be adapted to get a simple proof of an integrability theorem of Gromov and Eliashberg concerning higher-order nondegenerate immersions.     

Sharp estimates and a central limit theorem for the invariant law for a large star-shaped loss network

Calls arrive in a Poisson stream on a symmetric network constituted of N links of capacity C. Each call requires one channel on each of L distinct links chosen uniformly at random; if none of these links is full, the call is accepted and holds one channel per link for an exponential duration, else it is lost. The invariant law for the route occupation process has a semi-explicit expression similar to that for a Gibbs measure: it involves a combinatorial normalizing factor, the partition function, which is very difficult to evaluate. We study the large N limit while keeping the arrival rate per link fixed. We use the Laplace asymptotic method. We obtain the sharp asymptotics of the partition function, then the central limit theorem for the empirical measure of the occupancies of the links under the invariant law. We also obtain a sharp version for the large deviation principle proved in Graham and O'Connell (Ann. Appl. Probab. 10 (2000) 104).     

Meet-distributive lattices and the anti-exchange closure

This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that a latticeL is the lattice of closed sets of an anti-exchange closure if and only if it is a meet-distributive lattice. This result is used to give a combinatorial interpretation of the zetapolynomial of a meet-distributive lattice. Work done while the author was an Applied Mathematics Fellow at M.I.T. Presented by R. P. Dilworth     

Orbits of SU(2)-representations and minimal isometric immersions

In contrast to all known examples, we show that in the case of minimal isometric immersions of into the smallest target dimension is almost never achieved by an -equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of into . The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order play an important role in the study of the moduli space of all minimal isometric immersions of into . Using a new necessary and sufficient condition for immersions of isotropy order , we derive a general existence theorem of such immersions. Received: 13 May 1999 / in final form: 13 July 1999     

Existence and Uniqueness Theorem for Slant Immersions and Its Applications

A slant immersion is an isometric immersion from a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this paper we establish the existence and uniqueness theorem for slant immersions into complex-space-forms. By applying this result, we prove in this paper several existence and nonexistence theorems for slant immersions. In particular, we prove the existence theorems for slant surfaces with prescribed mean curvature or with prescribed Gaussian curvature. We also prove the non-existence theorem for flat minimal proper slant surfaces in non-flat complex space forms.     

On Possible Values of Upper and Lower Derivatives with Respect to Convex Differential Bases

It is proved that if a convex density-like differential basis B is centered and invariant with respect to translations and homotheties, then the integral means of a nonnegative integrable function with respect to B can boundedly diverge only on a set of measure zero (this generalizes a theorem of Guzmán and Menarguez); it is established that both translation and homothety invariances are necessary.     

On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.     

Immersions of graphs into projective plane

Immersions of graphs into the projective plane are studied. A classification of immersions up to a regular homotopy is obtained. A complete invariant of immersions up to a regular homotopy is constructed. The case of immersions of graphs into any compact surface differing from the projective plane was known previously.