Curvature weighted metrics on shape space of hypersurfaces in n-space
Curvature weighted metrics on shape space of hypersurfaces in n-space
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DOI：
10.1016/j.difgeo.2011.10.002
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发表时间：
2012-02-01
影响因子：
0.5
通讯作者：
Michor, Peter W.
中科院分区：
文献类型：
--
作者：
Bauer, Martin;Harms, Philipp;Michor, Peter W.
设M是一个无边界的紧连通定向（n - 1）维流形.在这项工作中，形状空间是从M到R“的非参数化浸入的轨道。M.作者声明：John W. Michor（2001）[1]研究了平均曲率加权度量，建议将高斯曲率权重纳入度量的定义中。这导致我们研究形状空间上的度量，这些度量由形式G（f）（h，k）= integral（M）Phi的浸入空间上的度量引起。(g)这里f是Imm（M，R“）的元素，是M到R”中的浸入，并且h，k是C-无穷大（M，R "）的元素，是f处的切向量。(g)其中，f *（g）over bar是R-n上的标准度量，j*（g）over bar是M上的诱导度量，vol（f*（g）over bar）是诱导体密度，Phi是一个依赖于平均曲率和高斯曲率的光滑函数.对于这些度量，我们计算的测地线方程的浸入空间和形状空间和保守的动量所产生的明显的对称性。数值实验说明了这些指标的行为。(C)2011 Elsevier B. V.保留所有权利。
Let M be a compact connected oriented (n - 1)-dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from M to R ''. The results of M. Bauer, P. Harms, P.W. Michor (2001) [1] where mean curvature weighted metrics were studied, suggest incorporating Gauss curvature weights in the definition of the metric. This leads us to study metrics on shape space that are induced by metrics on the space of immersions of the formG(f)(h, k) = integral(M) Phi.(g) over bar (h, k) vol (f*(g) over bar).Here f is an element of Imm(M, R '') is an immersion of M into R '' and h, k is an element of C-infinity(M, R '') are tangent vectors at f. (g) over bar is the standard metric on R-n, j*(g) over bar is the induced metric on M, vol(f*(g) over bar) is the induced volume density and Phi is a suitable smooth function depending on the mean curvature and Gauss curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space and the conserved momenta arising from the obvious symmetries. Numerical experiments illustrate the behavior of these metrics. (C) 2011 Elsevier B.V. All rights reserved.