Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds
The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism ac...
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| author | Mancho Manev |
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| description | The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon almost soliton, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. |
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| spelling | doaj-art-3b5ec57d061049e68bdf9fac7295f3472025-08-20T03:46:49ZengMDPI AGMathematics2227-73902025-06-011311186310.3390/math13111863Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian ManifoldsMancho Manev0Department of Algebra and Geometry, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen St., 4000 Plovdiv, BulgariaThe manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon almost soliton, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed.https://www.mdpi.com/2227-7390/13/11/1863<i>η</i>-Ricci–Bourguignon almost solitonalmost contact B-metric manifoldalmost contact complex Riemannian manifoldSasaki-like manifoldsolenoidal vector field |
| spellingShingle | Mancho Manev Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds Mathematics <i>η</i>-Ricci–Bourguignon almost soliton almost contact B-metric manifold almost contact complex Riemannian manifold Sasaki-like manifold solenoidal vector field |
| title | Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds |
| title_full | Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds |
| title_fullStr | Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds |
| title_full_unstemmed | Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds |
| title_short | Pair of Associated <i>η</i>-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds |
| title_sort | pair of associated i η i ricci bourguignon almost solitons with vertical potential on sasaki like almost contact complex riemannian manifolds |
| topic | <i>η</i>-Ricci–Bourguignon almost soliton almost contact B-metric manifold almost contact complex Riemannian manifold Sasaki-like manifold solenoidal vector field |
| url | https://www.mdpi.com/2227-7390/13/11/1863 |
| work_keys_str_mv | AT manchomanev pairofassociatediēiriccibourguignonalmostsolitonswithverticalpotentialonsasakilikealmostcontactcomplexriemannianmanifolds |