How a projectively flat geometry regulates F(R)-gravity theory?

In the present paper we examine a projectively flat spacetime solution of F(R)-gravity theory. It is seen that once we deploy projective flatness in the geometry of the spacetime, the matter field has constant energy density and isotropic pressure. We then make the condition weaker and discuss the effects of projectively harmonic spacetime geometry in F(R)-gravity theory and show that the spacetime in this case reduces to a generalised Robertson-Walker spacetime with a shear, vorticity, acceleration free perfect fluid with a specific form of expansion scalar presented in terms of the scale factor. Role of conharmonic curvature tensor in the spacetime geometry is also briefly discussed. Some analysis of the obtained results are conducted in terms of couple of F(R)-gravity models.

Projective flatness of a new class of ( α , β ) (\alpha,\beta) -metrics

In this paper we investigate a new {(\alpha,\beta)} -metric {F=\beta+\frac{a\alpha^{2}+\beta^{2}}{\alpha}} , where {\alpha=\sqrt{{a_{ij}y^{i}y^{j}}}} is a Riemannian metric; {\beta=b_{i}y^{i}} is a 1-form and {a\in(\frac{1}{4},+\infty)} is a real scalar. Also, we investigate the relationship between the geodesic coefficients of the metric F and the corresponding geodesic coefficients of the metric α.

PROJECTIVE STRUCTURES AND -CONNECTIONS

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

The Chern character of the Verlinde bundle over ℳ¯ g,n

We prove an explicit formula for the total Chern character of the Verlinde bundle of conformal blocks over \overline{\mathcal{M}}_{g,n} in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman’s classification of semisimple CohFTs, there exists an element of Givental’s group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the interior {\mathcal{M}}_{g,n} and the projective flatness of the Hitchin connection.

Riemannian manifolds satisfying certain conditions on pseudo-projective curvature tensor

In this paper we determine some properties of pseudo-projective curvature tensor denoted by ?P on some Riemannian manifolds, especially on generalized quasi Einstein manifolds in the sense of Chaki. Firstly, we consider a pseudo-projectively Ricci semisymmetric generalized quasi Einstein manifold. After that, we study pseudo-projective flatness of this manifold. Moreover, we construct a non-trivial example for a generalized quasi Einstein manifold to prove the existence.