THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

This paper studios the existence of closed geodesics in the homotopy class of a given closed curve. Let M be a complete Riemannian manifold without boundary, f∈C~1(S~1, M). Look at S~1 as 0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ(f_i), f_0=f always admits a global solution.B. (Existence of closed geodesics). If there exists a compact set KM such that f(S~1)∩K≠φ andE(f)≤(1/π)l(K)~2,then there exists a closed geodesic homotopic to f. If then the closed geodesic is minimal.C. Some estimates about injective radius are obtained.Some example is found showing that the inequalities in B are sharp.

THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

This paper studies the existence of closed geodesics in the homotopy class of a give closed curve. Let M be a complete Riemannian manifold without boundary, $\f \in {C^1}({S^1},M)\]$ Look at $\{S^1}\]$ as $\0,2\pi ]/\{ 0,2\pi \} \]$. The following results are proved: A. The initial value problem of heat equation $\{\partial _i}{f_i} = \tau ({f_i}),{f_0} = f\]$ always admits a global solution. B. (Existence of closed geodesics). If there exists a compact set $\K \subset M\]$ such that $\f({S^1}) \cap K \ne \phi \]$ and $$\E(f) \le \frac{1}{\pi }i{(\partial K)^2}\]$$ then there exists a closed geodesic homotopie to f. If $$\E(f) \le \frac{1}{\pi }i{(M\backslash K)^2}\]$$, then the closed geodesic is minimal. C. Some estimates abont injective radius are obtained. Some example is found showing that the inequalities in B are sharp.