arXiv research

Generalizations of the classical affine Lelieuvre formula to surfaces in projective three-dimensional space and to hypersurfaces in multi- dimensional projective space are given. A discrete version of the projective Lelieuvre formula is presented too.

The paper extends a formula linking surface curvature to projection invariants.

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

Formula for analytic torsion forms in fibrations by projective curves.

The paper develops formulas for hyperbolic simplices based on edge lengths.

For arbitrary quantizable compact Kaehler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realised via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vec…

Currents in higher dimensions can be reconstructed from projections.

We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossing divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invaria…

In [8], P. Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [1], M. Bordemann proved this existence using the framework of Thomas-Whitehead connections. In [9], we gave a new proof of the same theorem thanks to the Cartan connections. After these works there was no explicit f…

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit form…

The paper calculates delta invariants for specific geometric structures.

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the m…

Formula for projecting geodesics in hyperbolic 3-manifolds, relating lengths to subsurface projections.

The paper studies the correlation of Hilbert lengths for convex projective surfaces.

Formula counts rational curves with a specific singular point in projective space.

The paper finds transformation formulas for quaternionic complex structures.

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

Formula for Heisenberg group surface areas derived.

We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedr…

In [5], P. Lecomte conjectured the existence of a natural and conformally invariant quantization. In [7], we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the meth…

New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.

Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of $\mathbb{P}^n$ of codimension one. As a consequence …

A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power…

We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Lapl…

We prove residual formulas for vector fields defined on compact complex orbifolds with isolated singularities and give some applications of these on weighted projective spaces.

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…

Formula conjectured for rational cuspidal curves in projective plane.

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recu…

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can …

Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…

We give an alternative definition of relative hyperbolicity based on properties of closest-point projections on peripheral subgroups. We also derive a distance formula for relatively hyperbolic groups, similar to the one for mapping class groups.

We give a method to calculate spectra of the square of the Rarita-Schwinger operator on compact symmetric spaces. According to Weitzenböck formulas, the operator can be written by the Laplace operator, which is the Casimir operator on compact symmetric spaces. Then we can obtain the spectra by using the Freudenthal's f…

It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective.…

The paper derives a formula for Chow weights of toric blow-ups.

The paper studies geometric properties of Grassman manifolds within Euclidean spaces.

Let $\mathcal{G}^*(S,ρ)$ be the graph whose vertices are marked complex projective structures with holonomy $ρ$ and whose edges are graftings from one vertex to another. If $ρ$ is quasi-Fuchsian, a theorem of Goldman implies that $\mathcal{G}^*(S,ρ)$ is connected. If $ρ(π_1(S))$ is a Schottky group Baba has shown that …

In the paper we consider the theory of elliptic operators acting in subspaces defined by pseudodifferential projections. This theory on closed manifolds is connected with the theory of boundary value problems for operators violating Atiyah-Bott condition. We prove an index formula for elliptic operators in subspaces de…

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski t…

We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co-homogeneity one. Complete projective special real manifolds give rise to complete quate…

Formula derived for Gromov-Witten invariants of smooth curves.

Explicit formulas for Hattori-Stong integrability conditions and manifolds' signature properties.

Study Morse functions on projective plane using Reeb graphs.

We propose an explicit formula connecting Donaldson invariants and Seiberg-Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the N=2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when t…

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …

A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.