On Almost C(a)-Manifold Satisfying Some Conditions On The Weyl Projective Curvature Tensor

In the present paper, we have studied the curvature tensors of almost C()-manifolds satisfying the conditions P(,X)R = 0, P(,X) e Z = 0, P(,X)P = 0, P(,X)S = 0 and P(,X) e  C = 0. According these cases, we classified almost C()-manifolds

SOME CHARACTERIZATIONS OF α-COSYMPLECTIC MANIFOLDS ADMITTING ∗-CONFORMAL RICCI SOLITIONS

The object of the present paper is to give some characterizations of α-cosymplectic manifolds admitting ∗-conformal Ricci solitons. Such manifolds with gradient ∗-conformal Ricci solitons have also been considere

Sasaki-Einstein and paraSasaki-Einstein metrics from (\kappa,\mu)-structures

We prove that any non-Sasakian contact metric (\kappa,\mu)-space admits a canonical \eta-Einstein Sasakian or \eta-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of \kappa and \mu for which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (\kappa,\mu)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some topological and geometrical properties of (\kappa,\mu)-spaces related to the existence of Eistein-Weyl and Lorentzian Sasakian Einstein structures

ON THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS ADMITTING SCHOUTEN-VAN KAMPEN CONNECTION

In the present paper,  we study three-dimensional trans-Sasakian manifolds admitting the Schouten-van Kampen connection.  Also, we have proved some results on $\phi$-projectively flat, $\xi-$projectively flat and $\xi-$concircularly flat three-dimensional trans-Sasakian manifold  with respect to the Schouten-van Kampen connection. Locally $\phi-$symmetry trans-Sasakian manifolds of dimension three have been studied  with respect to Schouten-van Kampen connection. Finally, we construct an example of a three-dimensional trans-Sasakian manifold admitting Schouten-van Kampen connection which verifies Theorem 4.1

Geometrical aspects of contact mechanical systems and field theories

Many important theories in modern physics can be stated using the tools of differential geometry. It is well known that symplectic geometry is the natural framework to deal with autonomous Hamiltonian mechanics. This admits several generalizations for nonautonomous systems and classical field theories, both regular and singular. Some of these generalizations are the subject of the present dissertation. In recent years there has been a growing interest in studying dissipative mechanical systems from a geometric perspective by using contact structures. In the present thesis we review what has been done in this topic and go deeper, studying symmetries and dissipated quantities of contact systems, and developing the Lagrangian-Hamiltonian mixed formalism (Skinner-Rusk formalism) for these systems.With regard to classical field theory, we introduce the notion of k-precosymplectic manifold and use it to give a geometric description of singular nonautonomous field theories. We also devise a constraint algorithm for k-precosymplectic systems. Furthermore, field theories with damping are described through a modification of the De Donder-Weyl Hamiltonian field theory. This is achieved by combining both contact geometry and k-symplectic structures, resulting in what we call the k-contact formalism. We also introduce two notions of dissipation laws, generalizing the concept of dissipated quantity. The preceding developments are also applied to Lagrangian field theory. The Skinner-Rusk formulation for k-contact systems is described in full detail and we show how to recover both the Lagrangian and Hamiltonian formalisms from it. Throughout the thesis we have worked out several examples both in mechanics and field theory. The most remarkable mechanical examples are the damped harmonic oscillator, the motion in a constant gravitational field with friction, the parachute equation and the damped simple pendulum. On the other hand, in field theory, we have studied the damped vibrating string, the Burgers' equation, the Klein-Gordon equation and its relation with the telegrapher's equation, and the Maxwell's equations of electromagnetism with dissipation.Moltes teories de la física moderna es poden formular mitjançant les eines de la geometria diferencial. Com és ben sabut, la geometria simplèctica és el marc natural per treballar amb sistemes mecànics Hamiltonians autònoms. La geometría simplèctica admet diverses generalitzacions que permeten treballar amb sistemes no autònoms i amb teories de camps, tant en el cas regular com en el singular. En aquesta tesi treballarem amb algunes d'aquestes generalitzacions. En els darrers anys l'interés per l'estudi geomètric dels sistemes mecànics dissipatius mitjançant estructures de contacte ha crescut notablement. En aquesta tesi revisem el que s'ha fet sobre aquest tema i anem més enllà, estudiant les seves simetries i quantitats dissipades, i desenvolupant el formalisme mixt lagrangià-hamiltonià (formalisme de Skinner-Rusk) per aquests sistemes. En referència a la teoria de camps, introduïm la noció de varietat k-precosimplèctica i la usem per a descriure geomètricament les teories de camps no autònomes singulars. També desenvolupem un algorisme de lligams per a sistemes k-precosimplèctics. A més, descrivim les teories de camps amb dissipació mitjançant una modificació de la teoria hamiltoniana de camps de De Donder-Weyl. Aquesta descripció combina la geometria de contacte i les estructures k-simplèctiques, obtenint el que anomenem formalisme de k-contacte. També generalitzem el concepte de quantitat dissipada introduint dues nocions de llei de dissipació. Aquests desenvolupaments s'apliquen també a les teories de camps lagrangianes. Donem una descripció completa de la formulació de Skinner-Rusk dels sistemes de k-contacte i veiem com recuperar els formalismes lagrangià i hamiltonià a partir del formalisme mixt. Al llarg de la tesi hem estudiat diversos exemples tant en mecànica com en teoria de camps. Els sistemes mecànics més rellevants són l'oscil·lador harmònic esmorteït, el moviment en un camps gravitatori constant amb fregament, l'equació del paracaigudista i el pèndul simple esmorteït. Per altra banda, en teoria de camps estudiem la corda vibrant esmorteïda, l'equació de Burgers, l'equació de Klein-Gordon i la seva relació amb l'equació del telegrafista, i les equacions de Maxwell de l'electromagnetisme amb dissipació.Postprint (published version