Compact Riemannian Manifolds with Homogeneous Geodesics

A homogeneous Riemannian space $(M= G/H,g)$ is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group $G$. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric $g$ with homogeneous geodesics on a homogeneous space of a compact Lie group $G$. We give a classification of compact simply connected GO-spaces $(M = G/H,g)$ of positive Euler characteristic. If the group $G$ is simple and the metric $g$ does not come from a bi-invariant metric of $G$, then $M$ is one of the flag manifolds $M_1=SO(2n+1)/U(n)$ or $M_2= Sp(n)/U(1)\cdot Sp(n-1)$ and $g$ is any invariant metric on $M$ which depends on two real parameters. In both cases, there exists unique (up to a scaling) symmetric metric $g_0$ such that $(M,g_0)$ is the symmetric space $M = SO(2n+2)/U(n+1)$ or, respectively, $\mathbb{C}P^{2n-1}$. The manifolds $M_1$, $M_2$ are weakly symmetric spaces

Kinetic equations for ultrarelativistic particles in a Robertson-Walker Universe and isotropization of relict radiation by gravitational interactions

Kinetic equations for ultrarelativistic particles with due account of gravitational interactions with massive particles in the Robertson-Walker universe are obtained. On the basis of an exact solution of the kinetic equations thus obtained, a conclusion is made as to the high degree of the uniformity of the relict radiation on scales with are less than $10'$.Comment: 19 pages, 2 figures, 13 reference