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Characterizing Round Spheres Using Half-Geodesics
A half-geodesic is a closed geodesic realizing the distance between any pair
of its points. All geodesics in a round sphere are half-geodesics. Conversely,
this note establishes that Riemannian spheres with all geodesics closed and
sufficiently many half-geodesics are round
Homogeneous geodesics in homogeneous Finsler spaces
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces.
We first give a simple criterion that characterizes geodesic vectors. We show
that the geodesics on a Lie group, relative to a bi-invariant Finsler metric,
are the cosets of the one-parameter subgroups. The existence of infinitely many
homogeneous geodesics on compact semi-simple Lie group is established. We
introduce the notion of naturally reductive homogeneous Finsler space. As a
special case, we study homogeneous geodesics in homogeneous Randers spaces.
Finally, we study some curvature properties of homogeneous geodesics. In
particular, we prove that the S-curvature vanishes along the homogeneous
geodesics
Geodesic measures of the landscape
We study the landscape models of eternal inflation with an arbitrary number
of different vacua states, both recyclable and terminal. We calculate the
abundances of bubbles following different geodesics. We show that the results
obtained from generic time-like geodesics have undesirable dependence on
initial conditions. In contrast, the predictions extracted from ``eternal''
geodesics, which never enter terminal vacua, do not suffer from this problem.
We derive measure equations for ensembles of geodesics and discuss possible
interpretations of initial conditions in eternal inflation.Comment: 7 pages, 4 figure
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