We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...

We construct infinitely many two-dimensional Finsler metrics on S 2 and D 2 with non-zero constant flag curvature. They are all not locally projectively flat.

In this paper we analyzed the problem of investigating locally the scalar curvature of the two dimensional kinematic surfaces foliated by the homothetic motion of an eight curve in LorentzMinkowski 5-space . We express the scalar curvature of the corresponding two dimensional kinematic surfaces as the quotient of hyperbolic functions { }. From that point, we derive the necessary and sufficient ...

Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N . In particular, we give sufficient conditions, involving π1(N) and dim N , for N to admit an infinite number of metrics of positive scalar curvature that...

the general relatively isotropic mean landsberg metrics contain the general relativelyisotropic landsberg metrics. a class of finsler metrics is given, in which the mentioned two conceptsare equivalent. in this paper, an interpretation of general relatively isotropic mean landsberg metrics isfound by using c-conformal transformations. some necessary conditions for a general relativelyisotropic ...

The state space of a finite quantum system is identified with the set of positive semidefinite matrices of trace 1. The set of all strictly positive definite matrices of trace 1 becomes naturally a differentiable manifold and the Bogoliubov-Kubo-Mori scalar product defines a Riemannian structure on it. Reference [4] tells about the relation of this metric to the von Neumann entropy functional. ...

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of g genus with g > 2 does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which is equivalent to the Teichmüller metric, on any finite cover of the moduli space is ...