Canonical connections on paracontact manifolds
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In this article it was shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the structure is skew symmetric and the defining vector field is Killing.Abstract:
The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$
-homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$
-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.read more
Citations
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Contact geometry and thermodynamics
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Contact symmetries and Hamiltonian thermodynamics
TL;DR: In this paper, the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics is investigated. And the authors use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics.
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On Legendre Curves in 3-Dimensional Normal Almost Paracontact Metric Manifolds
TL;DR: In this article, the curvature and torsion properties of Legendre curves in 3-dimensional normal almost paracontact metric manifolds are studied, and properties of non-Frenet Legendre curve (with null tangents or null normals or null binormals) are obtained.
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Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry
TL;DR: In this paper, the authors studied the class of parallel symmetric tensor fields of dimension 3 and possible Lorentz Ricci solitons for paracontact geometry of dimension three.
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Nullity conditions in paracontact geometry
TL;DR: In this paper, a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tilde \kappa$ and $\tilde\mu$) is presented.
References
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Book
Riemannian Geometry of Contact and Symplectic Manifolds
TL;DR: In this article, the authors describe a complex geometry model of Symplectic Manifolds with principal S1-bundles and Tangent Sphere Bundles, as well as a negative Xi-sectional Curvature.
Book
Contact manifolds in Riemannian geometry
TL;DR: In this paper, the tangent sphere bundle is shown to be a contact manifold, and the contact condition is interpreted in terms of contact condition and k-contact and sasakian structures.
Journal ArticleDOI
Parallel spinors and connections with skew-symmetric torsion in string theory
Thomas Friedrich,Stefan Ivanov +1 more
TL;DR: In this article, all almost contact metric, almost hermitian and G2-structures admit a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection.
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Variational problems on contact Riemannian manifolds
TL;DR: In this paper, the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds was defined, and the torsion and generalized Tanaka-Webster scalar curvature were defined properly.