arXiv research

Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.

Stable approach solves equivariant Hopf theorem for G-manifolds.

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…

We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincaré-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology…

New tool: relative Hopf invariant for Poincaré surgery.

We give a construction of cyclic cocycles representing the equivariant characteristic classes of equivariant bundles. Our formulas generalize Connes' Godbillon-Vey cyclic cocycle. An essential tool of our construction is Connes-Moscovici's theory of cyclic cohomology of Hopf algebras.

The paper studies coherent sheaves on subvarieties of Hopf manifolds.

The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy c…

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…

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 geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…

Classifies Real primary Hopf surfaces and their associated groups.

Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments,lacking a microscopic underpinning in M-brane charge quantizatio…

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…

Generalizes Hopf degree theorem to nontrivial bundles.

Global inverse function theorem proved easily using Riemannian geometry.

We show that the S^1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S^1-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S^1-actio…

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

New invariants derived from link homology for 4-manifolds.

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 Jänich presented a Poincaré-Hopf th…

Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

The Hopf fibration is rigid among minimal maps between spheres.

We survey generalisations of the Chang-Skjelbred Lemma for integral coefficients. Moreover, we construct examples of manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday's. The "mutants" we construct are obtained from compactified repre…

The Hopf-Rinow theorem is extended to sub-Finslerian geometry.

Hopf's theorem generalized to curved spaces.

New minimal hypersurfaces in 4D sphere found.

A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.

In this paper we give an extension of the Cartier-Gabriel-Kostant structure theorem to Hopf algebroids.

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

Holomorphic vector bundles on Hopf manifolds admit flat connections.

This paper deals with a semi-classical limit (Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussion of it may be found in [7].

We state and prove a generalization of the Poincaré-Hopf index theorem for manifolds with boundary. We then apply this result to non-vanishing complex vector fields.

In this paper we construct a new family of harmonic morphisms $\varphi:V^5\to\s^2$, where $V^5$ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of $\c^4=\r^8$. These harmonic morphisms admit a continuous extension to the completion ${V^{\ast}}^5$, which turns out to be an explicit real algebra…

The paper extends Hopf's theorem to convex surfaces and discrete triangulations.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in…

Proves an equivariant version of index theorem for geometric families.

Extends E. Hopf's theorem to magnetic systems without conjugate points.

Symmetric hypersurfaces with constant mean curvature are spheres.

In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem for spin manifolds without boundary. For $6$ dimensional spin manifolds with boundary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then we generalize this theorem to the general $n$ dimensional manifold. An equivariant Kastler-Kal…

New proofs for complex Hopf manifolds using geometric structures.

Researchers prove an equivariant index theorem on Euclidean space.

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant K…

Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.

Extends Tian theorem to Vaisman manifolds for approximations.