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In this paper, the idea of the Ricci flow is introduced and its significance and importance to related problems in mathematics had been discussed. Several functionals are defined and their behavior is studied under Ricci flow. A unique minimizer is shown to exist for one of the functionals. This functional evaluated at the minimizer is strictly increasing. The results for the first functional considered are extended to manifold with boundary. Finally, two physically motivated examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study f-biharmonic Riemannian submersion, bi-f-harmonic Riemannian submersion and thus generalizing some results of [M. A. Akyol and Y.-L. Ou, Biharmonic Riemannian submersions, Ann. Math. Pura Appl.198 (2019) 559–570; Z. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z.269(3) (2011) 917–925]. Next, we obtain the conditions when the Riemannian submersion on its first factor is f-biharmonic as well as bi-f-harmonic. In the last section, we study f-biharmonic cylinders of a Riemannian submersion. [ABSTRACT FROM AUTHOR]

Let M be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that M is η -parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in ℂ P 2 or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in ℂ H 2 . Moreover, η -parallel real hypersurfaces in ℂ P 2 and ℂ H 2 under some other conditions are classified and these results extend Suh's in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo's in [On characterizations of real hypersurfaces in a complex space form with η -parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126]. [ABSTRACT FROM AUTHOR]

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish some general Kastler–Kalau–Walze type theorems for any dimensional manifolds with boundary which generalize the results in [J. Wang and Y. Wang, The Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary, J. Math. Phys.56 (2015), Article ID: 052501, 14 pp.]. [ABSTRACT FROM AUTHOR]

In this paper, -inflation is analyzed in warm braneworld scenario. A general class of -essence models with power-law kinetic term is investigated and weak and strong dissipation regimes are studied. Scalar perturbations and spectral index are derived. The results are discussed and applied to specific examples. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of Levi-Civita connections for strongly σ -compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in [J. Bhowmick, D. Goswami and S. Mukhopadhyay, Levi-Civita connections for a class of spectral triples, Lett. Math. Phys. 110 (2020) 835–884] and [J. Bhowmick, D. Goswami and G. Landi, On the Koszul formula in noncommutative geometry, Rev. Math. Phys. 32(10) (2020) 2050032]. [ABSTRACT FROM AUTHOR]

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated "events", associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser.968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states. [ABSTRACT FROM AUTHOR]

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen. 39(33) (2006) 10449- 10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145-154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a 3-form and a 2-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background. [ABSTRACT FROM AUTHOR]

The cosmological dynamics of a non-locally corrected gravity theory, involving a power of the inverse d'Alembertian, is investigated. Casting the dynamical equations into local form, the fixed points of the models are derived, as well as corresponding de Sitter and power-law solutions. Necessary and sufficient conditions on the model parameters for the existence of de Sitter solutions are obtained. The possible existence of power-law solutions is investigated, and it is proven that models with de Sitter solutions have no power-law solutions. A model is found, which allows to describe the matter-dominated phase of the Universe evolution. [ABSTRACT FROM AUTHOR]

The Cotton tensor of three-dimensional Walker manifolds is investigated. A complete description of all locally conformally flat Walker three-manifolds is given, as well as that of Walker manifolds whose Cotton tensor is either a Codazzi or a Killing tensor. [ABSTRACT FROM AUTHOR]