Paul Kinlaw
Dartmouth College
| A complete Riemannian manifold (M,g) is called a Y^x_l-manifold if every geodesic c parametrized by arc length with c(0)=x satisfies c(l)=x from some x in M and l>0. Bérard-Bergery's theorem includes the following result: a Y^x_l-manifold of dimension >1 is compact and has finite fundamental group. We call (M,g) a Y^x-manifold if for every epsilon>0 there exists l>epsilon such that for every geodesic c parametrized by arc length with c(0)=x, we have d(c(l),x)<epsilon. We will cover recent work including a proof that Y^x-manifolds of dimension >1 are compact with finite fundamental group. We will discuss the relationship between Y^x_l and Y^x-manifolds and refocusing Lorentz manifolds. |
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