Real hypersurfaces in complex space forms

In this thesis the problem of classifying three dimensional real hypersurfaces in complex projective space form CP2 and in complex hyperbolic space form CH2, in terms of conditions concerning the structure Jacobi operator ℓ has been studied. In Chapter 1 basic notations and lemmas of the theory of differentiable manifolds and submanifolds are included. In Chapter 2 the three dimensional real hypersurfaces in complex projective space CP2 and in complex hyperbolic space CH2, additional basic notations and relations, which hold on them, are presented. In Chapter 3 the classification of three dimensional real hypersurfaces, whose structure Jacobi operator ℓ is ξ-parallel, i.e. ξ ℓ=0, where ξ is the covariant derivative with respect to the structure vector field ξ, which is defined on real hypersurfaces, is given. In Chapter 4 the notion of Lie D-parallelness is introduced the non-existence of real hypersurfaces with Lie D-parallel structure Jacobi operator is proved. In Chapter 5 three dimensional real hypersurfaces equipped with structure Jacobi operator satisfying the relation Lξℓ= ξℓ, where Lξ is the Lie derivative with respect to the structure vector field ξ, is given. The notation of pseudo-parallel structure Jacobi operator is introduced and real hypersurfaces with this structure Jacobi operator ℓ are classified. Finally, in Chapter 7 the classification of three dimensional real hypersurfaces with cyclic-parallel structure Jacobi ℓ is presented.Το αντικείμενο μελέτης της παρούσας διατριβής είναι η ταξινόμηση των τρισδιάστατων πραγματικών υπερεπιφανειών μέσα στο μιγαδικό προβολικό χώρο CP2 και στο μιγαδικό υπερβολικό χώρο CH2, όταν ο τελεστής δομής Jacobi ℓ αυτών ικανοποιεί ορισμένες συνθήκες. Πιο αναλυτικά, στο πρώτο κεφάλαιο περιέχονται βασικές έννοιες και λήμματα από τη θεωρία των διαφορισίμων πολλαπλοτήτων και υποπολλαπλοτήτων. Στο δεύτερο κεφάλαιο αναφέρονται γνωστά αποτελέσματα και αποδεικνύονται σχέσεις που ισχύουν σε οποιεσδήποτε πραγματικές υπερεπιφάνειες των CP2 και CH2. Στο τρίτο κεφάλαιο ταξινομούνται οι τρισδιάστατες πραγματικές υπερεπιφάνειες, των οποίων ο τελεστής δομής Jacobi ℓ είναι ξ-παράλληλος, δηλαδή ξℓ=0, όπου ξ είναι η συναλλοίωτος παράγωγος κατά τη κατεύθυνση του διανυσματικού πεδίου δομής ξ που ορίζεται σε μια πραγματική υπερεπιφάνεια. Στο τέταρτο κεφάλαιο ορίζεται πότε η Lie παράγωγος του τελεστή δομής Jacobi είναι D-παράλληλη και αποδεικνύεται ότι τέτοιου είδους τρισδιάστατες πραγματικές υπερεπιφάνειες δεν υπάρχουν. Στο πέμπτο κεφάλαιο ταξινομούνται οι τρισδιάστατες πραγματικές υπερεπιφάνειες των οποίων ο τελεστής δομής Jacobi ℓ ικανοποιεί τη σχέση Lξℓ= ξℓ, όπου Lξ είναι η Lie παράγωγος κατά τη κατεύθυνση του διανυσματικού πεδίου δομής ξ. Στο έκτο κεφάλαιο αναφέρεται η έννοια της ψευδο-παραλληλίας ενός τανυστικού πεδίου και επιπλέον ταξινομούνται οι τρισδιάστατες πραγματικές υπερεπιφάνειες των οποίων ο τελεστής δομής Jacobi ℓ είναι ψευδο-παράλληλος. Τέλος, στο έβδομο κεφάλαιο γίνεται ταξινόμηση των τρισδιάστατων πραγματικών υπερεπιφανειών με κυκλικά παράλληλο τελεστή δομής Jacobi ℓ

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.SCOPUS: ar.jSCOPUS: ar.jinfo:eu-repo/semantics/publishe

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations

Quest for Bioactive Compounds in Our Diet with Anti-Ageing and Anti-Aggregation Properties

Ageing is a complex process affected by both genetic and environmental factors, characterized by a gradual failure of functionality, reduced stress response and resistance, leading to enhanced probability for age-related diseases and mortality. During the last decades, natural compounds have attracted the attention of researchers in the quest of bioactive phytochemicals with anti-ageing properties. For a few of these compounds an extra advantage appears; many of them have been shown to decelerate the progression of age-related diseases with emphasis on aggregation-related diseases. Using the nematode Caenorhabditis elegans along with the replicative senescence model of human primary fibroblasts, we have identified compounds that are part of our diet with anti-oxidation, anti-ageing and anti-aggregation activities. Some of the identified compounds promote their anti-ageing activity through activation of the proteasome, others through the activation of Nrf2 transcription factor, while others through inhibition of glucose transporters (GLUTs). Our work identifies new bioactive compounds with anti-ageing and/or anti-aggregation properties or reveals additional beneficial properties on already known bioactive compounds

A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

The authors would like to thank the reviewers for their valuable comments in order to improve the paper.The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F (k) X is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by FX and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that FXS = SFX, where S denotes the Ricci tensor of M and a further condition is satisfied, are classified