arXiv research

Summarizes connections between Euler characteristic theorems and conjectures.

Proves a generalized table theorem for odd Euler characteristic surfaces.

Euler derived elastica equation using modern mathematical concepts.

Euler's theorem extended to complex structures.

Study Euler class of surface bundles with nontrivial results.

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…

Study Loday algebroids, prove splitting theorem, and linearize problems.

Extends Euler class formula to general connections with metric.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDiff^δ_ω(S^1);\mathbb{Q})$ is zero where $Diff^δ_ω(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler clas…

Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisf…

The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…

The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two diffe…

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a neg…

A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler c…

The paper investigates the relationship between curvature operator and Euler number on manifolds.

Paper studies critical points of curvature energies in 4D.

Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric structures to the construction of an Euler-like vector field compatible with the given …

The paper proves that certain stationary hypersurfaces in high dimensions are essentially flat.

Formula proves Euler characteristic of singularized surfaces.

This paper generalizes Batchelor's theorem in $C^\infty$-superschemes.

Despite the fact that the Euler allocation principle has been adopted by many financial institutions for their internal capital allocation process, a comprehensive description of Euler allocation seems still to be missing. We try to fill this gap by presenting the theoretical background as well as practical aspects. In…

We prove a comparison theorem for the compact surfaces with negative Euler characteristic via the Ricci flow.

We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ vanishes unless $A=TM$, and prove a general Künneth formula. As appl…

Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.

In the note, we give a proof, based on the Generalized Thom Conjecture, of Bennequin's Theorem on upper bound for the Euler number of a link which is considered as a closed braid. A lower bound for the Euler number of a link is also given.

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data…

Using the equivalence between the renormalized Euler characteristic of Ozsvath and Szabo, and the Turaev torsion normalized by the Casson-Walker invariant, we make calculations for $S^3_{p/q}(K)$. An alternative proof of a theorem by Ozsváth and Szabó on $L$-space surgery obstructions is provided.

In this study, some characterizations of Euler spirals in E_1^{3} have been presented by using their main property that their curvatures are linear. Moreover, discussing some properties of Bertrand curves and helices, the relationship between these special curves in E_1^{3} have been investigated with different theorem…

We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.

Proves Massey's theorems on complex structure obstructions.

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat A(TM)e(F),[M]>=0$, where $e(F)$ is the Euler class of $F$.

We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.

Study vector fields on non-compact manifolds with group action.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.

A formula calculates the Euler class of foliations using dual graphs.

Toolbox for stochastic Euler equations using Ebin-Marsden theory.

The study identifies unique fluid flow patterns.

Improved theorem on curvature and manifold symmetry.

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincaré--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corres…

The goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem and the index of the vec…

Geometric framework for dissipative systems on Lie algebroids.

In [2], N.Dutertre and T. Fukui used Viro's integral calculus to study the topology of stable maps $f:M\rightarrow N$ between two smooth manifolds $M$ and $N$. They also discussed several applications to Morin maps. In particular, in Theorem 6.2 [2], they show an equality relating the Euler characteristic of a compact …

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

We construct an analogue of the classical theta-function on an Abelian variety for closed 4-dimensional symplectic manifolds which are T^2-bundles over T^2 with the zero Euler class. We use our theta-functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analogue of the Lef…

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.