The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry.Inpainting is realized by completing damaged isophotes(level lines of brightness)b...The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry.Inpainting is realized by completing damaged isophotes(level lines of brightness)by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations(motions)of a plane SE(2).The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision.A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed.The approach and the algorithm for computation of completing arcs are presented in detail.展开更多
Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension c...Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests.展开更多
In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give ...In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.展开更多
In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg gro...In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.展开更多
In this paper,we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C^(2)-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and s...In this paper,we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C^(2)-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C^(2)-smooth curves on surfaces.We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.展开更多
In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carn...In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.展开更多
The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions...The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincard lemma stated on l：tiemannian manifolds and a suitable Ceskro-Volterra path in- tegral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.展开更多
In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the ...In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.展开更多
基金The authors acknowledge support by Russian Foundation for Basic Research,Project No.12-01-00913-aand by the Ministry of Education and Science of Russia within the Federal program"Scientific and Scientific-Pedagogical Personnel of Innovative Russia,"Agreement No.8209 of August 6,2012.
文摘The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry.Inpainting is realized by completing damaged isophotes(level lines of brightness)by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations(motions)of a plane SE(2).The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision.A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed.The approach and the algorithm for computation of completing arcs are presented in detail.
基金the National Natural Science Foundation of China(Grant No.11771303)the second author was also partially supported by the Beijing Advanced Innovation Center for Imaging Theory and Technology,Capital Normal University.
文摘Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests.
文摘In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.
文摘In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.
基金supported by National Natural Science Foundation of China(Grant No.11771070)。
文摘In this paper,we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C^(2)-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C^(2)-smooth curves on surfaces.We get Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane.
基金supported by the National Natural Science Foundation of China(No.10471063)
文摘In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.
文摘The author proves the Poincard lemma on some （n ＋1）-dimensional corank 1 sub-Riemannian structures, formulating the （n-1）n（n2＋3n-2） necessarily and sufficient- s ly ＂curl-vanishing＂ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincard lemma stated on l：tiemannian manifolds and a suitable Ceskro-Volterra path in- tegral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.
基金supported by National Security Agency,United States Army Research Offfice and a Hong Kong RGC Competitive Earmarked Research (Grant No. 600607)
文摘In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ? n+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.
