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Summary: CURVE SHORTENING AND THE TOPOLOGY OF CLOSED
GEODESICS ON SURFACES
SIGURD B. ANGENENT
Abstract. We study \
at knot types" of geodesics on compact surfaces M 2 .
For every
at knot type and any Riemannian metric g we introduce a Conley
index associated with the Curve Shortening
ow on the space of immersed
curves on M 2 . We conclude existence of closed geodesics with prescribed
at
knot types, provided the associated Conley index is nontrivial.
1. Introduction
If M is a surface with a Riemannian metric g then closed geodesics on (M; g)
are critical points of the length functional L(
) =
R
j
0 (x)jdx de ned on the space
of unparametrized C 2 immersed curves with orientation, i.e. we consider closed
geodesics to be elements of the
space
= Imm(S 1 ; M)=Di + (S 1 ):
Here Imm(S 1 ; M) = f
2 C 2 (S 1 ; M) j
0 () 6= 0 for all 2 S 1 g and Di + (S 1 ) is
the group of C 2 orientation preserving di eomorphisms of S 1 = R=Z. (We will
abuse notation freely, and use the same symbol
to denote both a convenient
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