Invited Speakers

Title: $(M_{+},M_{-})$-theory and its applications to topology, groups, geometry and combinatorics

Abstract:

Maximal antipodal sets on a compact Riemannian manifold were introduced by Chen and Nagano in 1982. Since then a number of mathematicians had studied maximal antipodal sets with many nice applications/connections to several areas of mathematics. In this talk, I will present a brief survey on applications of maximal antipodal set

Title: A proof of the Toponogov conjecture 

Abstract:

In 1995 Toponogov authored the following conjecture: “Every smooth strictly convex and complete surface of the type of a plane has an umbilic point, possibly at infinity“. In our talk, we will outline a proof, in collaboration with Brendan Guilfoyle, namely that (i) the Fredholm index of an Riemann Hilbert boundary problem for holomorphic discs associated to a putative counterexample is negative. Thereby, (ii) no solutions may exist for a generic perturbation of the boundary condition (iii) however, the geometrization by a neutral metric gives rise to barriers for the continuity method to prove existence if a holomorphic disc.

Title: On semi-Riemannian manifolds satisfying some conformally invariant curvature condition

Abstract: Let C be the Weyl conformal curvature tensor of a semi-Riemannian manifold (M,g), dim M = n > 3, and let U the set of all points of M at which C is nonzero. The manifold (M,g) is said to have a pseudosymmetric conformal Weyl tensor if the (0,6)-tensors $C \cdot C$ and $Q(g,C)$, formed by the metric tensor g and the Weyl tensor C, are linearly dependent at every point of M, i.e., if $C\cdot C = L Q(g,C)$ on U, where L is some function on U. Semi-Riemannian manifold (M,g), dim M > 3, satisfying this condition is also called a manifold with pseudosymmetric Weyl tensor.

For instance, the following non-conformally flat warped product manifolds are manifolds with pseudosymmetric Weyl tensor:

warped product manifolds with a 1-dimensional base and a 3-dimensional quasi-Einstein fiber,

warped product manifolds with a 1-dimensional base and an (n-1)-dimensional quasi-Einstein conformally flat fiber, n > 4,

warped product manifolds with a 2-dimensional base and a 2-dimensional fiber,

and warped product manifolds with a 2-dimensional base and an (n-2)-dimensional fiber, n > 4, assumed that the fiber is a space of constant curvature.

The Schwarzschild spacetime, the Kottler spacetime, the Reissner-Nordstroem spacetime, the Goedel spacetime and Roter spaces,

as well as 2-quasi-umbilical hypersurfaces isometrically immersed in spaces of constant curvature, are such manifolds (hypersurfaces).

In this talk we present results on manifolds (hypersurfaces) with pseudosymmetric Weyl tensor satisfying some additional curvature conditions.

This is a joint work with Malgorzata Glogowska and Jan Jelowicki.

Title: Riemannian Submersions

Abstract: We discuss here, various classes of Riemannian submersions, including holomorphic submersions, invariant submersions, anti-invariant submersions, semi-invariant submersions, Generic submersions, slant submersions, semi-slant submersions, hemi-slant submersions, pointwise slant and semi-slant submersions, bi-slant and quasi-bi-slant submersions, conformal submersions, etc. We also discuss the integrability conditions and the geometry of leaves of the distributions that are involved in the definition of submersions. Further, We focus on some necessary and sufficient conditions for these submersions to be totally geodesic, and harmonic. Some examples, and applications of the submersions are also discussed.

Title: Why We Work in Higher Dimensions?

Abstract: With the help of tensor calculus and a little bit of general relativity, a genuine physical answer has been found to the very basic question that "why the higher dimensions are necessary to work for", while only the tree dimensional are visible.

Title: On the class of anti-quasi-Sasakian manifolds

Abstract: I will discuss some special classes of almost contact metric manifolds such that the transverse geometry with respect to the 1-dimensional foliation generated by characteristic vector field is given by a Kähler structure. I will focus on quasi-Sasakian manifolds and on the new class of anti-quasi-Sasakian manifolds. In this case, the transverse geometry is given by a Kähler structure endowed with a closed 2-form of type (2,0), as for instance hyperkähler structures. I will describe examples of anti-quasi-Sasakian manifolds, including compact nilmanifolds and principal circle bundles, investigate Riemannian curvature properties, and the existence of connections with torsion preserving the structure. This is a joint work with D. Di Pinto.

Title: 3-dimensional almost contact metric manifolds with a new perspective

Abstract:

The geometrical objects, functions, vector fields, 1-forms, and in general, tensors on any manifold have an important role in differential geometry, especially in the construction of structures on manifolds.  Many works have focused on three-dimensional almost contact metric structures, either by studying and classifying them or by employing them to study various topics. The aim of this talk is two-fold. First, an interesting expression that generalizes all classes of 3-dimensional almost contact metric structures. Second, A new method to construct all almost contact metric structures on a 3-dimensional Riemannian manifold starting from a unit vector field $\xi$ . By this approach, we give a technique to discuss the nature of such structures.

Title: Geometric Inequalities Involving $\delta-$Casorati Curvatures: A Survey

Abstract: In order to explore the relationship between intrinsic and extrinsic invariants, Chen [1] established a sharp inequality to derive Chen's invariants, also referred to as $\delta-$-invariants. This finding made it possible to discover a new field of differential geometry and explore Chen-type and Chen invariant inequalities for submanifolds in different ambient spaces. The work of Casorati [2] made it possible to replace the conventional Gauss curvature with the Casorati curvature. Decu et al. [3] initiated the investigation of $\delta-$- Casorati curvatures in the context of $\delta-$-invariants. The present study offers a thorough analysis of recent studies conducted over the last few years on $\delta-$-Casorati curvatures.

 References

[1] Chen, B.Y., Some pinching and classification theorems for minimal submanifolds, Arch. Math., 60 (1993), 568–578.

[2] Casorati, F., Mesure de la courbure des surfaces suivant l’id´ee commune, Acta Math., 14 (1890), 95–110.

[3] Decu, S., Haesen, S., Verstraelen, L., Optimal inequalities involving Casorati curvatures, Bull. Transylv. Univ. Brasov, Ser. B, 14 (2007), 85–93.