Differentiation in metric spaces
Differentiation in metric spaces
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DOI：
10.1090/s1061-0022-05-00888-5
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发表时间：
2005
影响因子：
0.8
通讯作者：
A. Lytchak
中科院分区：
文献类型：
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作者：
A. Lytchak
1.1.目标。本文主要研究度量空间的一阶几何。我们的研究主要是出于这样的观察，即虽然具有上曲率界和下曲率界的亚历山德罗夫空间理论的高级特征是完全不同的，但开始几乎是相同的，至少就一阶导数而言（例如切空间和第一变分公式）。一个自然是导致的问题，在空间上的一阶几何可以建立。事实证明，同样的一阶几何也存在于许多我们称之为几何的其他空间中。这类几何空间包含所有的保持器连续黎曼流形，充分凸和光滑的Finsler流形（[LY]），黎曼流形的一大类子集（例如正到达集，见[Fed59]和[Lyta]），具有积分曲率界的曲面（[Res93]）和具有较低曲率界的Alexandrov空间的极值子集（[PP94 a]）。最后一种情况在[Pet94]中进行了讨论，第一个变分公式的证明是证明深度胶合定理（[Pet94]）的重要一步。此外，这类几何空间在度量运算下是稳定的，即使在像取逆元这样困难的运算下也是如此。最后，一阶几何的存在性是研究高阶特征的一个很好的假设，例如半凹函数的梯度流（[PP94b]和[Lytc]）。本文的主要问题之一是建立自然的，易于验证的公理，描述这种一阶几何及其后果。
1.1. The Aim. This paper is devoted to the study of the first order geometry of metric spaces. Our study was mainly motivated by the observation that whereas the advanced features of the theories of Alexandrov spaces with upper and lower curvature bounds are quite different, the beginnings are almost identical, at least as far as only first order derivatives are concerned (for example tangent spaces and the first variation formula). One is naturally led to the question on which spaces the first order geometry can be established. As it turns out the same first order geometry exists in many other spaces that we call geometric. The class of geometric spaces contains all Holder continuous Riemannian manifolds, sufficiently convex and smooth Finsler manifolds ([LY]), a big class of subsets of Riemannian manifolds (for example sets of positive reach, see [Fed59] and [Lyta]), surfaces with an integral curvature bound ([Res93]) and extremal subsets of Alexandrov spaces with lower curvature bound ([PP94a]). The last case was discussed in [Pet94] and the proof of the first variation formula was a major step towards proving the deep gluing theorem ([Pet94]). Moreover the class of geometric spaces is stable under metric operations, even under such a difficult one as taking quotients. Finally the existence of the first order geometry is a good assumption for studying features of higher order, such as gradient flows of semi-concave functions ([PP94b] and [Lytc]). One of the main issues of this paper is the establishing of natural, easily verifiable axioms, that describe this first order geometry and their consequences.