Related papers: Amalgamated Codazzi Raychaudhuri identity for foli…
Embedding fields provide a way of coupling a background structure to a theory while preserving diffeomorphism-invariance. Examples of such background structures include embedded submanifolds, such as branes; boundaries of local subregions,…
It is shown that the timelike, spacelike and null versions of the Ehlers identity, as well as ensuing Raychaudhuri equations, might be all derived within a single geometrical approach based on the definition of the Riemann curvature tensor…
We show that the covariant Raychaudhuri identity describing kinematic characteristics of space-time admits a representation involving a geometrical scalar $\xi$ which, depending on circumstances, might be related to, e.g., relativistic…
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic…
We present a geometric framework for reconstruction problems based on Vaisman foliations and Atiyah--Molino sequences. Independent projections induce transverse foliations and dual connections; vanishing torsion and curvature duality…
The Ricci curvature equations are a central subject of study in geometry. However, in the smooth real case, their linear analysis is often confined to settings in which the background metric is Einstein. In this paper, we establish…
A decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of constant sectional curvature. In particular, it is shown that if $M$ has nonnegative sectional curvature and admits a Codazzi…
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…
In this paper we study the problem of analytic extension of germs of holonomy of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is…
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally…
We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic…
We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincar\'e duality in the transversally orientable case. We use this twisted basic…
This work deals with the topological classification of singular foliation germs on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the…
We introduce a local vector field on an $n$-dimensional Riemannian manifold, defined as the sum of the covariant derivatives of a local orthonormal frame, and derive an explicit identity for its divergence, decomposed into a scalar…
On the basis of the generalizations of the Jacobi identity found by the author some identities satisfied by the curvature and torsion of a covariant differentiation are derived. A kind of the generalized covariant differentiation is…
In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation…
We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the…
We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface,…
The questions of global topological, smooth and holomorphic classifications of the differential systems, defined by covering foliations, are considered. The received results are applied to nonautonomous linear differential systems and…
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give…