A class ofU(n)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres

1993 ◽  
Vol 34 (4) ◽  
pp. 612-619 ◽  
Author(s):  
V. N. Berestovskii ◽  
D. E. Vol'per
Keyword(s):  
Riemannian Metrics ◽  
Invariant Riemannian Metrics

scholarly journals Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups

2013 ◽  
Vol 1 ◽  
pp. 255-275 ◽  
Author(s):  
Luca Capogna ◽  
Giovanna Citti ◽  
Maria Manfredini
Keyword(s):  
Total Variation ◽  
Carnot Group ◽  
Riemannian Metrics ◽  
Heat Kernels ◽  
Gradient Estimates ◽  
Long Time Existence ◽  
Total Variation Flow ◽  
Long Time ◽  
Invariant Riemannian Metrics ◽  
Time Existence

Abstract In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σε ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ε → 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ε = 0), which in turn yield sub-Riemannian minimal surfaces as t → ∞.

scholarly journals On invariant Riemannian metrics on Ledger–Obata spaces

2018 ◽  
Vol 158 (3-4) ◽  
pp. 353-370 ◽  
Author(s):  
Y. Nikolayevsky ◽  
Yu. G. Nikonorov
Keyword(s):  
Riemannian Metrics ◽  
Invariant Riemannian Metrics

scholarly journals Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

2016 ◽  
Vol 68 (2) ◽  
pp. 669-684 ◽  
Author(s):  
Takahiro HASHINAGA ◽  
Hiroshi TAMARU ◽  
Kazuhiro TERADA
Keyword(s):  
Lie Groups ◽  
Riemannian Metrics ◽  
Invariant Riemannian Metrics

scholarly journals Elastica in SO(3)

2007 ◽  
Vol 83 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Tomasz Popiel ◽  
Lyle Noakes
Keyword(s):  
Riemannian Manifold ◽  
Computer Graphics ◽  
Variational Problem ◽  
Lie Groups ◽  
Real Line ◽  
Lagrange Equation ◽  
Riemannian Metrics ◽  
Geodesic Curvature ◽  
Unit Speed ◽  
Invariant Riemannian Metrics

AbstractIn a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

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