Related papers: A Poincar\'e-Hopf theorem for n-valued vector fiel…
In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any…
In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.
In this work, we present an alternative definition of the Le Roux index, which generalizes the Poincar\'e-Hopf index for non-singular planar flows to the broader setting of Brouwer homeomorphisms. This new approach answers a question raised…
In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar…
In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two…
We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle $E$ of even rank over a closed compact orientable manifold $M$. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special…
We have shown that two of the most studied models of lineal gravities - Liouville gravity and a ``string-inspired'' model exhibiting the main characteristic features of a black-hole solution - can be formulated as gauge invariant theories…
The quantum hyperplane section theorem is explained for nonnegative decomposable concavex bundle spaces over generalized flag manifolds.
If (V,0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V one can define an index of X, the so called GSV index, which generalizes the Poincare-Hopf index. We prove that the GSV index coincides…
A map between manifolds which matches up families of complete vector fields is a fiber bundle mapping on each orbit of those vector fields.
We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. By utilizing a vector field induced isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product,…
Over an algebraically closed field k, there are 16 lines on a degree 4 del Pezzo surface, but for other fields the situation is more subtle. In order to improve enumerative results over perfect fields, Kass and Wickelgren introduce a method…
The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz…
We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by…
It is shown that certain lower semi-continuous maps from a paracompact space to the family of closed subsets of the bundle space of a Banach bundle admit continuous selections. This generalization of the theorem of Douady, dal…
It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincar\'e-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$…
In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value…
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds.…
Given a locally injective real function g on the vertex set V of a finite simple graph G=(V,E), we prove the Poincare-Hopf formula f_G(t) = 1+t sum_{x in V} f_{S_g(x)}(t), where S_g(x) = { y in S(x), g(y) less than g(x) } and f_G(t)=1+f_0 t…
We give an proof on the Weinstein conjecture on the cotangent bundles of open manifolds. Its proof is based on Gromov's nonlinear Fredholm alternative.