arXiv research

4D Poincaré complexes have simpler fibrations.

Research shows reducibility of low dimensional Poincaré complexes in certain cases.

Contradicts claims about Poincaré complexes and homology manifolds.

Study homology manifolds using spectral sheaves and spectral six functor formalism.

New method normalizes Milnor fibrations for real analytic maps.

Study normal operators of double fibration transforms with conjugate points.

This paper explores the relationship between generalized manifolds and Poincaré duality complexes.

Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras o…

In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constru…

This paper corrects errors in UMAP's derivation and explains its properties.

The paper explores properties of shape m$_{ m simpl}$-fibrators among Hopfian manifolds.

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a general response of a problem started with an id…

Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…

Formula connects analytic torsion forms of fibration and its pieces.

In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurabili…

In this paper, helicoidal flat surfaces in the $3$-dimensional sphere $\mathbb{S}^3$ are considered. A complete classification of such surfaces is given in terms of their first and second fundamental forms and by linear solutions of the corresponding angle function. The classification is obtained by using the Bianchi-S…

Study shows scalar curvature of a specific type of manifold converges to -m outside singular points.

Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartan's theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the s…

In this article we obtain a result about the uniqueness of factorization in terms of conjugates of the matrix $U=(\xymatrix{1 & 1 0 & 1})$, of some matrices representing the conjugacy classes of those elements of $SL(2,Z)$ arising as the monodromy around a singular fiber in an elliptic fibration (i.e. those matrices th…

We study the existence of projectable $G$-invariant Einstein metrics on the total space of $G$-equivariant fibrations $M=G/L\to G/K$, for a compact connected semisimple Lie group $G$. We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators, which is a generali…

Random high-dimensional manifolds lack totally geodesic submanifolds.

Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.

The article studies mapping properties of Radon transform and backprojection on a unit ball.

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the a…

We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's Theorem on product of almost contact manifolds to flat bundles. We construct som…

Study projective structures and rational curves to understand Painlevé equations.

Study X-ray transform on manifolds, desingularize, and improve mapping properties.

Paper shows a geometry construction has universal limits property.

In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms …

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is not transitive (on t…

Study harmonicity of normal almost contact structures on Riemannian manifolds.

We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they carry an almost complex structure defined using the horizontal distribution of t…

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsev…

The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, th…

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous sp…

The study proves that certain surface singularities are planar only for $A_n$-singularities.

Study local curvature of Kähler-Ricci flow on semi-ample manifolds.

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\geq 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrati…

This thesis studies geometric structures using Lie algebroids to simplify singular behavior.

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

The paper generalizes Hannay-Berry connections for foliated manifolds with symmetry.

New Poisson and near-symplectic structures found on 6D wrinkled fibrations.

Introduces new stability concept for Fano fibrations.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

Study of line fibrations in R^3, focusing on non-skew cases.

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.