RIEMANNIAN CENTER OF MASS AND MOLLIFIER SMOOTHING
RIEMANNIAN CENTER OF MASS AND MOLLIFIER SMOOTHING
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DOI：
10.1002/cpa.3160300502
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发表时间：
1977-01-01
影响因子：
3
通讯作者：
KARCHER, H
中科院分区：
文献类型：
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作者：
KARCHER, H
很明显，如何将经典的软化平滑（窄核卷积）推广到从黎曼流形到线性空间的映射。在第三节中，我们利用文[6]中引入的质心把光滑化推广到黎曼流形之间的映射。该结构非常适合于度量，以至于关于近似或关于Lipschitz估计的保持的标准结果继续存在（第4节）。第五节给出了双Lipschitz映射的光滑性具有最大秩的一个充分条件。Shikata定理是一个直接的推论;事实上，我们得到了比以前已知的更清晰的版本。我们已经作出了特别的努力，使文件可读的人没有太多的背景微分几何。特别是，分析师应该首先阅读刚才提到的第3节和第5节。我们从第一节开始对质心进行简化处理，并在第二节中描述说明其用途的一些应用。此外，微分几何的论点并没有分散在整个文件，但收集在三个附录。在附录A中，我们证明了所有需要的Jacobi估计。我们需要已知的和新的估计;已知的是证明在某种程度上广义的形式。我们提请注意推论A5和命题A6，其中包含精确的信息的导数和旋转的雅可比场。在附录B中，我们简要地解释了如何用雅可比场来描述几何构造。附录C结合了附录A和B来证明众所周知的弧长比较定理（见C2）的推广，比较切空间中的平行平移与黎曼平行平移（见C2）。2)并估计不经常使用的黎曼指数映射的导数（在C3中）。关于这项工作，我从与J.J.B.B.的讨论中获益匪浅，并与特别研究领域的下列客人进行了讨论：Shikata，K.盐滨湾格罗夫，HC Im霍夫。从1966年到1968年，我在纽约大学柯朗研究所度过了两个富有成效的年头;回想起来，这种影响在我的工作中是显而易见的。我感谢所有这些
It is rather obvious how to generalize the classical mollifier smoothing (convolution with narrow kernels) to maps from Riemannian manifolds into lineur spaces. We use the center of mass which was introduced in [6] to generalize the mollifier smoothing to maps between Riemannian manifolds in Section 3. The construction is so well adapted to the metric that the standard results about approximation or about preservation of Lipschitz estimates carry over (Section 4). In Section 5 we give a sufficient condition under which the smoothing of a bi-Lipschitz map has maximal rank. Shikata’s theorem is an immediate corollary; in fact we obtain a sharper version than was previously known. We have made a special effort to make the paper readable for people with not too much background in differential geometry. In particular, analysts should read the just mentioned Sections 3 and 5 first. We start in Section 1 with a simplified treatment of the center of mass and describe in Section 2 some applications which illustrate its use. Also, the differential-geometric arguments are not scattered through the paper but collected in three appendices. In Appendix A we prove all the Jacobi estimates which are needed. We need known and new estimates; the known ones are proved in a somewhat generalized form. We draw attention to Corollary A5 and Proposition A6 which contain precise information about the derivative and the rotation of Jacobi fields. In Appendix B we briefly explain how Jacobi fields are used to describe geometric constructions. Appendix C combines Appendices A and B to prove a generalization of well known arc length comparison theorems (see C2), to compare parallel translation in the tangent space with Riemannian parallel translation (in C2. 2) and to estimate not so frequently used derivatives of the Riemannian exponential map (in C3). In connection with this work I profited from discussions with J. Eschenburg and with the following guests of the Sonderforschungsbereich: Y. Shikata, K. Shiohama, K. Grove, HC Im Hof. Between 1966 and 1968 I spent two fruitful years at the Courant Institute of New York University; in retrospect that influence is clear in my work. I am grateful for all those