On the number of solutions for the two-point boundary value problem on Riemannian manifolds
- 作者：Masiello ; Antonio ; Piccione ; Paolo
- 关键词：Differential geometry ; Dynamical system ; 26B25 ; 34B15 ; 52A41 ; 58E05 ; 58E10 ; Two-point boundary value problem
- 刊名：Journal of Geometry and Physics
- 出版年：2004
- 期刊代码：29_03930440
- 类别：ph
- 出版时间：January, 2004
- 卷：49
- 期：1
- 页码：67-88
- 文件大小：190 K
摘要
We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M,g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p,q
M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.
M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.