Cosmological meaning of geometric curvatures

Cosmological Geometric Phase

It is shown that a wave vector representing a light pulse in an adia-batically evolving expanding space should develop, after a round trip (back and forth to the emitter) a geometric phase for helicity states at a given fixed position coordinate of this expanding space.In a section of the Hopf fibration of the Pioncaré sphere S 2 that identifies a projection to the physically allowed states, th...

Estimation of Curvatures in Point Sets based on Geometric Algebra

Gaussian Curvature * 1 Geometric Interpretation of Principal Curvatures

We have learned that the two principal curvatures (and vectors) determine the local shape of a point on a surface. One characterizes the rate of maximum bending of the surface and the tangent direction in which it occurs, while the other characterizes the rate and tangent direction of minimum bending. The rate of surface bending along any tangent direction at the same point is determined by the...

Lower bounds on Ricci flow invariant curvatures and geometric applications

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+ε I ∈ C at the initial time, then it satisfies R+Kε I ∈ C on some time interval depen...

Quantum Geometric Description of Cosmological Models

This is a written version of the review talk given at the meeting on " Interface of Gravitational and Quantum Realms " at IUCAA, Pune during December 2001. The talk reviewed the recent work of Martin Bojowald on Loop Quantum Cosmology.