کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4606298 | 1337696 | 2012 | 9 صفحه PDF | دانلود رایگان |
Let M be a compact connected oriented (n−1n−1)-dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from M to RnRn. 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 formGf(h,k)=∫MΦ.g¯(h,k)vol(f⁎g¯). Here f∈Imm(M,Rn)f∈Imm(M,Rn) is an immersion of M into RnRn and h,k∈C∞(M,Rn)h,k∈C∞(M,Rn) are tangent vectors at f . g¯ is the standard metric on RnRn, f⁎g¯ is the induced metric on M , vol(f⁎g¯) is the induced volume density and Φ 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.
Journal: Differential Geometry and its Applications - Volume 30, Issue 1, February 2012, Pages 33–41