Searcharxiv⌕ Search

arXiv subjects

Mihail Cocos

Publications and source records attributed to Mihail Cocos.

9 recordsLinked to original sources
arXiv2025

Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections

We study a class of affine manifolds equipped with a flat affine connection $\nabla$ and a global Riemannian metric $g$ that is diagonal in local affine coordinates. These structures are closely related to \emph{Hessian manifolds}, where the metric locally arises as the Hessian of a smooth potential. For example, the Hopf manifold $(\mathbb{R}^{n+1}\setminus \{0\}) / \langle x \mapsto 2x \rangle$ with metric $g = (\sum_i x_i^2)^{-1} \sum_i dx_i^2$ admits a proper deformation of $\nabla$ into its Levi-Civita connection. By Theorem 2.3 in \cite{cocos2025}, such deformations force the Euler characteristic to vanish, providing evidence for Chern's conjecture. The geometry of these manifolds is reminiscent of the work of Yau on affine and Hessian structures \cite{cheng_yau_1986}.

math.DG↗
arXiv2025

A Riemannian Characterization of Compact Affine Manifolds with Parallel Volume

We establish that any affine manifold $(M,\nabla)$ endowed with a parallel volume form $\omega,$ admits, in any conformal class of Riemannian metrics, a representative $H$ for which $\nabla$ is the Levi-Civita connection. This provides a constructive proof that such manifolds are necessarily complete, generalizing the "if" direction of Markus' conjecture \cite{markus1962}. Moreover, our result demonstrates that these structures are intrinsically Riemannian-flat, a stronger conclusion than the affine completeness asserted by Markus. The metric $H$ arises naturally from the Hessian of volume-normalized distance functions and is shown to be globally smooth and $\nabla$-parallel, extending results of \cite{goldman1982} and \cite{benzecri1955} to higher dimensions with additional geometric structure. The construction proceeds through three technically novel steps: (1) local parallel metric normalization using the given volume form, (2) explicit Hessian calculations in adapted coordinates, and (3) gluing via affine transition maps that preserve the volumetric geometry. This approach reveals an unexpected rigidity in flat affine manifolds with compatible volume that goes beyond the topological constraints studied in \cite{fried1980}.

math.DG↗
arXiv2016

A note on locally metric connections

In this paper we give necessary and sufficient conditions for a connection in a plane bundle above a surface to be locally metric. These conditions are easy to be verified in any local chart. Also as a global result we give a necessary condition for two connections to be metric equivalent in terms of their Euler class.

math.DG↗
arXiv2016

Skewable matrices over $\wedge^2 V$ applied to locally metric connections

A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of $2$ forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of $2$ forms is equivalent to a skew symmetric matrix. We apply this algorithm to verify whether a "{\it full rank}" curvature connection is locally metric.

math.DG↗
arXiv2012

Zero sets and factorization of polynomials of two variables

The relationship between a polynomial's zeros and factors is well known. If a is a zero of f(x) then (x-a) is a factor of f(x). In this paper, we generalize this idea to polynomials of two variables and with real coefficients. We consider the zero sets of two variable polynomials and give criterion to when two polynomials with the same zero set have a common factor with the same zero set. When the coefficients of the polynomials are not in a field, but the division algebra of Quaternions, we provide an example of two polynomials with the same zero set and no common factor.

math.AG↗
arXiv2012

On the topology of compact affine manifolds

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption is dropped, the manifold is not necessarily obtained as the quotient of the Euclidean space through a properly discontinuous group of affine transformations. In fact the universal cover may no longer be the Euclidean space. The main result of this paper states that all compact affine manifolds have 0 Euler characteristic and that the fundamental group of these manifolds is non-trivial.

math.DG↗
arXiv2012

Musical Modes, Their Associated Chords and Their Musicality

In this paper we present a mathematical way of defining musical modes and we define the musicality of a mode as a product of three different factors. We conclude by classifying the modes which are most musical according to our definition.

math.CO↗
arXiv2011

Music By Numbers

In this paper we present a mathematical way of defining musical modes, we derive a formula for the total number of modes and define the musicality of a mode as the total number of harmonic chords whithin the mode. We also give an algorithm for the construction of a duet of melodic lines given a sequence of numbers and a mode. We attach the .mus files of the counterpoints obtained by using the sequence of primes and several musical modes.

math.CO↗