相关论文: Regular homotopy classes of immersions of 3-manifo…

Totally real immersions $f$ of a closed real surface $\Sigma$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings from…

Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times…

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism…

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an embedded template in $S^3$ with fixed homeomorphism type, its boundary as a…

We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a…

In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R^{2n-1} to locally generic mappings. The main result is that if n=2 then two mappings with singularities are regularly homotopic…

We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect…

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation-equivalent to…

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy…

We define an invariant of triple-point-free immersions of $2$-spheres into Euclidean $3$-space, taking values in $l^1(\mathbb{Z})$. It remains unchanged under regular homotopies through such immersions. An explicit description of its image…

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the…

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of…

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature…

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an…

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic…

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x: M \to R^3…

For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We…

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for…

Let f: M -> N be an even codimensional immersion between smooth manifolds. We derive an explicit formula for the Pontrjagin numbers and signature of the multiple point manifolds in terms of singular cohomology of M and N, the maps induced…

Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the somewhat intricate case when the surface is nonorientable.