THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

【摘要】 <正> 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∈C¹（S¹, M）. Look at S¹ as [0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ（f_i）, f₀=f always admits a global solution.B. （Existence of closed geodesics）. If there exists a compact set KM such that f（S¹）∩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.更多还原

【Abstract】 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∈C¹（S¹, M）. Look at S¹ as [0, 2π]/{0, 2π}. The following results are proved: A. The initial value problem of heat equation _if_t=τ（f_i）, f₀=f always admits a global solution. B. （Existence of closed geodesics）. If there exists a compact set KM such that f（S¹）∩K≠φ and E（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.更多还原