Related papers: Gauss-Bonnet type theorems in any dimension

First order invariants of generic immersions of manifolds of dimension nm-1 into manifolds of dimension n(m+1)-1, m,n>1 are constructed using the geometry of self-intersections. The range of one of these invariants is related to Bernoulli…

We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where…

The Gauss-Bonnet theorem for a polyhedron (a union of finitely many compact convex polytopes) in $n$-dimensional Euclidean space expresses the Euler characteristic of the polyhedron as a sum of certain curvatures, which are different from…

In this paper we study the rational homotopy of the space of immersions, $Imm\left(M,N\right)$, of a manifold $M$ of dimension $m\geq 0$ into a manifold $N$ of dimension $m+k$, with $k\geq 2$. In the special case when $N=\mathbb{R}^{m+k}$…

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new…

The Gauss map of a generic immersion of a smooth, oriented surface into $\mathbb R^4$ is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in $\mathbb R^4$. Since this manifold has a structure of a product of…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we…

In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of…

We classify immersions $f$ of $S^1$ in a $2$-manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the number of double points of a self-transverse $C^1$-approximation of $f$, and the turning number $T(e\bar f)$ of the…

We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different…

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and…

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically…

Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We…

The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical…

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of…

In their paper "Integrating curvature: From Umlaufsatz to J+ invariant" Lanzat and Polyak introduced a polynomial invariant of generic curves in the plane as a quantization of Hopf's Umlaufsatz, and showed that Arnold's J+ invariant could…

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb R^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$…

For a compact differentiable surface with boundary embedded in $\Bbb R^3$, we give simple proofs of the Gauss-Bonnet theorem, Poincar\'{e}-Hopf theorem, and several other integral formulas. We complete all of the proofs without using…

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…