Characterization of Ricci Almost Soliton on Lorentzian Manifolds

Abstract

Ricci solitons (RS) have an extensive background in modern physics and are extensively used in cosmology and general relativity. The focus of this work is to investigate Ricci almost solitons (RAS) on Lorentzian manifolds with a special metric connection called a semi-symmetric metric u-connection (SSM-connection). First, we show that any quasi-Einstein Lorentzian manifold having a SSM-connection, whose metric is RS, is Einstein manifold. A similar conclusion also holds for a Lorentzian manifold with SSM-connection admitting RS whose soliton vector Z is parallel to the vector u. Finally, we examine the gradient Ricci almost soliton (GRAS) on Lorentzian manifold admitting SSM-connection.

On the Geometry of Kobayashi–Nomizu Type and Yano Type Connections on the Tangent Bundle with Sasaki Metric

“…and ( ∇ 1 kn , ∇ 2 y ), where ∇, ∇ 1 and ∇ 2 are torsion-free connections on M. In our future work, we will explore how the main results in this paper can be applied in conjunction with soliton theory, submanifold theory, and other related fields (as discussed in [12][13][14][15][16][17][18][19][20][21][22][23]) to yield more new results.…”

Quasi-Statistical Schouten–van Kampen Connections on the Tangent Bundle

“…By using (40) and then (37) and (38), we obtain that the numerators of A 4 (t) and A 6 (t) become, respectively:…”

“…and it vanishes, according to Lemma 1, if and only if the involved coefficients vanish simultaneously. Since the metric G is proper general natural, i.e., c 3 = 0, d 3 = 0, the coefficient of g 0i g 0j g 0k vanishes if and only if one of the following instances happens: (I.1.i) c = 0, which together with (37), (38), (40), (43), (44) leads to:…”

“…, for every κ 3 ∈ R \ {0}, t > 0, but together with (38), (40), (37), and (43), which would imply d 2 = − c 2 2t , i.e., c 2 + 2td 2 = 0 and d 3 = c 3 , i.e., c 3 + 2td 3 = 0, and hence, the second nondegeneracy condition (6) of the metric G would not be satisfied.…”

“…The new structures are, thus, uncorrelated with the ones from the base, therefore constituting a more convenient geometric setting to investigate the statistical behavior in depth. Thus, new opportunities are opened for applications in information theory, machine learning, neural networks, statistical mechanics and geometry of Ricci solitons, for which we cite [38][39][40][41] and the references therein.…”