Tensor Analysis for Physicists. By J.A. Schouten. Second edition. Pp. xii, 277. 30s. 1954. (Geoffrey Cumberlege, Oxford University Press.)

Tangent and cotangent bundles : differential geometry

Differential geometry on complex and almost complex spaces

In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.

Contact Hamiltonian systems

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.

A survey on cosymplectic geometry

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKahler manifolds. We also mention some of their applications to time-dependent mechanics.

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a Survey on Cosymplectic Geometry

We prove that on a compact Sasakian manifold $(M,\eta, g)$ of dimension $2n + 1$, for any $0 \leq p \leq n$ the wedge product with $\eta \wedge (d \eta)^p$ defines an isomorphism between the spaces of harmonic forms $\Omega^{n-p}_{\Delta}(M)$ and $\Omega^{n+p+1}_{\Delta}(M)$. Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p}(M)$ and $H^{n+p+1}(M)$. Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.

/pdf/hard-lefschetz-theorem-for-sasakian-manifolds-52d5etzzlp.pdf

Hard Lefschetz Theorem for Sasakian manifolds

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

In the present article we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular, we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.

/pdf/3-quasi-sasakian-manifolds-43l6ah1t7y.pdf

Antonio De Nicola

Papers

A survey on cosymplectic geometry

a Survey on Cosymplectic Geometry

Hard Lefschetz Theorem for Sasakian manifolds

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

3-Quasi-Sasakian manifolds