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16 pages, 253 KiB

Open AccessArticle

\tilde{J}

-Tangent Affine Hypersurfaces with an Induced Almost Paracontact Structure

by Zuzanna Szancer

Symmetry 2025, 17(6), 806; https://doi.org/10.3390/sym17060806 - 22 May 2025

Viewed by 310

Abstract

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on [...] Read more.

The subjects of our study are affine hypersurfaces

f : M \to R^{2 n + 2}

considered with a transversal vector field C, which is

\tilde{J}

-tangent. By

\tilde{J}

we understand the canonical paracomplex structure on

R^{2 n + 2}

. The vector field C induces on the hypersurface f an almost paracontact structure

(φ, ξ, η)

. We obtain a complete classification of hypersurfaces admitting a metric induced almost paracontact structure with respect to the second fundamental form. We show that, in this case, the

\tilde{J}

-tangent transversal vector field is restricted to centroaffine and the hypersurface must be a piece of hyperquadric. It is demonstrated that these hyperquadrics have a very specific form. A three-dimensional example is also given. Moreover, we establish an equivalence relation between almost paracontact metric structures, para

α

-contact metric structures, and para

α

-Sasakian structures. Methods of affine differential geometry, as well as paracomplex/paracontact geometry, are used. Full article

(This article belongs to the Section Mathematics)

10 pages, 283 KiB

Open AccessArticle

Para-Ricci-Like Solitons on Riemannian Manifolds with Almost Paracontact Structure and Almost Paracomplex Structure

by Hristo Manev and Mancho Manev

Mathematics 2021, 9(14), 1704; https://doi.org/10.3390/math9141704 - 20 Jul 2021

Cited by 9 | Viewed by 2115

Abstract

We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements. Full article

(This article belongs to the Section B: Geometry and Topology)

19 pages, 237 KiB

Open AccessArticle

Conformal Gauge Transformations in Thermodynamics

by Alessandro Bravetti, Cesar S Lopez-Monsalvo and Francisco Nettel

Entropy 2015, 17(9), 6150-6168; https://doi.org/10.3390/e17096150 - 2 Sep 2015

Cited by 15 | Viewed by 5531

Abstract

In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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