Almost Local Metrics on Shape Space of Hypersurfaces in n-Space
Almost Local Metrics on Shape Space of Hypersurfaces in n-Space
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DOI：
10.1137/100807983
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发表时间：
2012-01-01
影响因子：
2.1
通讯作者：
Michor, Peter W.
中科院分区：
文献类型：
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作者：
Bauer, Martin;Harms, Philipp;Michor, Peter W.
本文推广了[P.W.Michor和D.Mumford，Appl.电脑。哈蒙。分析，23(2007)，pp.74-113]，将平面曲线推广到R-n中的超曲面的情况。设M是紧致连通定向的n维流形，无边界，类似于球面或环面。则形空间是M型R-n子流形的流形，或者是模M的微分同胚群的从M到Rn的浸入的分支。我们研究了形空间上的几乎局部黎曼度量。这些度量是由浸入空间上的如下度量导出的：G(F)(h，k)-积分(M)Phi(Vol(F)，Tr(L))(G)over bar(h，k)Vol(f*(G)over bar)，其中(G)over bar是R-n上的欧几里得度量，f*(G)over bar是M，h，k上的诱导度量。C-无穷大(M，R-n)是f处与嵌入或浸入空间的切向量，其中F：R-2>；R>；0是一个合适的光滑函数，Vol(F)=Vol(M)Vol(f*(G)on bar)是f(M)的全超曲面体积，Weingarten映射的迹Tr(L)是平均曲率。对于这些度规，我们计算了浸入空间和形状空间上的测地线方程、由明显对称性产生的守恒动量和截面曲率。对于F的特殊选择，我们给出了完整的截面曲率公式。数值实验说明了这些度量的行为。
This paper extends parts of the results from [P. W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74-113] for plane curves to the case of hypersurfaces in R-n. Let M be a compact connected oriented n - 1 dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of R-n of type M or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: G(f)(h, k) - integral(M) Phi(Vol(f), Tr(L))(g) over bar (h, k) vol(f* (g) over bar), where (g) over bar is the Euclidean metric on R-n, f* (g) over bar is the induced metric on M, h, k. C-infinity(M, R-n) are tangent vectors at f to the space of embeddings or immersions, where F : R-2 -> R->0 is a suitable smooth function, Vol(f) = integral(M) vol(f* (g) over bar) is the total hypersurface volume of f(M), and the trace Tr(L) of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of F we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.