Rigidity for surfaces of non-positive curvature
Rigidity for surfaces of non-positive curvature
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非正曲率曲面的刚度
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发表时间：
1990
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影响因子：
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通讯作者：
C. Croke
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作者：
C. Croke
本文研究非正曲率曲面的边界刚性问题。给定一个紧致流形M，其边界N是光滑的，M上的黎曼度量go在N × N上导出一个非负的真实的值函数do，其中do（p，q）是p和q在（M，go）中的距离.称黎曼流形（M，go）为边界刚性的，如果对任意具有相同边界N的黎曼流形（M~，g~），若dt = do，则gt等距于go.这个问题是最近考虑的作者在[C]其中一个是导致question：“是所有SGM流形边界刚性？".条件SGM是关于边界距离函数do的条件，粗略地说，它等价于M中的所有测地线段都是端点之间的唯一最小化路径的条件（参见[C]的精确定义）。通过测地线段，我们指的是最多在边界点处与边界相交的测地线（即，它们不在段的内部点处“掠过”边界）。凸流形内部的任何可能具有空边界（即任何两点之间存在唯一测地线）的紧致子域都将是SGM。因此，特别地，非正曲率的完备单连通流形的任何子域都是SGM。任何非正曲率的圆盘也是SGM。在本文中，我们显示：
In this paper we consider the question of boundary rigidity for surfaces of nonpositive curvature. Given a compact manifold, M, with smooth boundary N, a riemannian metric go on M induces a nonnegative real valued function, do, on N x N where do(p, q) is the distance in (M, go) between p and q. A riemannian manifold (M, go) is called boundary rigid if for any riemannian manifold (M~, g~) with the same boundary, N, if d t = do then gt is isometric to go. This question was recently considered by the author in [C] where one was led to the quesiton: "Are all SGM manifolds boundary rigid?". The condition SGM is a condition on the boundary distance function do which roughly speaking is equivalent to the condition that all geodesic segments in M are the unique minimizing paths between the endpoints (see [C] for a precise definition.) By geodesic segments we mean geodesics that intersect the boundary at most at the boundary points (i.e. they do not "graze" the boundary at interior points of the segment.) Any compact subdomain in the interior of a convex manifold with possibly empty boundary (i.e. between any two points there is a unique geodesic) will be SGM. Hence, in particular, any subdomain of a complete simply connected manifold of nonpositive curvature will be SGM. Also any disk of nonpositive curvature will be SGM. In this paper we show: