The Experts below are selected from a list of 69 Experts worldwide ranked by ideXlab platform
Domenico Perrone - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Vector Fields
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent Vector Field on a Riemannian manifold, first and second variation formulae, and the Harmonic Vector Fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where Harmonic Vector Fields occur and to generalizations. Any unit Vector Field that is a Harmonic map is also a Harmonic Vector Field. The study of Harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian Harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.
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Geometry of the Tangent Bundle
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary This chapter briefly reviews the basic facts in the geometry of the tangent bundle over a Riemannian manifold, such as nonlinear connections, the Dombrowski map, and the Sasaki metric. Tangent bundle also carries a natural almost complex structure compatible to Sasaki metric and such that tangent bundle and complex structure are almost Kahler manifold. The results offer the possibility of further development for the Harmonic Vector Field theory. A deep circle of ideas relates the geometry of the tangent bundle over a Riemannian manifold to the study of the global solutions to the homogeneous complex Monge-Ampere equation. It further examines the behavior of the total bending functional under conformational transformation. Conclusive and complete results on Harmonic Vector Fields on Riemannian tori are presented.
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Chapter Five – Harmonicity with Respect to g-Natural Metrics
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary The Harmonic map equations for unit Vector Fields are discussed for the particular case of Reeb Fields in contact metric geometry. It is always a Harmonic Vector Field and it is a Harmonic map under appropriate conditions on the coefficients determining the fixed g-natural metric, allowing one to exhibit large families of Harmonic maps defined on a compact Riemannian manifold and having a target space with a highly nontrivial geometry. Vertical Harmonicity with respect to Riemannian g-natural metrics appears to be worth further investigation. In the theory of Harmonic maps, a fundamental question concerns the existence of Harmonic maps between two given Riemannian manifolds. It is therefore important to find examples of Harmonic maps into Riemannian manifolds whose sectional curvature is not necessarily non-positive.
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Instability of the geodesic flow for the energy functional
Pacific Journal of Mathematics, 2011Co-Authors: Domenico PerroneAbstract:Let (S n (r),g 0 ) be the canonical sphere of radius r. Denote by G s the Sasaki metric on the unit tangent bundle T 1 S n (r) induced from g 0 and by G s the Sasaki metric on T 1 T 1 S n (r) induced from G s . We resolve here, for n ≥ 7, a question raised by Boeckx, Gonzalez-Davila, and Vanhecke: namely, we prove that the geodesic flow ξ: (T 1 S n (r), G s ) → (T 1 T 1 S n (r), G s ) is an unstable Harmonic Vector Field for any r > 0 and n ≥ 7. In particular, in the case r = 1, ξ is an unstable Harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric G s .
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Stability of the Reeb Vector Field of H-contact manifolds
Mathematische Zeitschrift, 2008Co-Authors: Domenico PerroneAbstract:It is well known that a Hopf Vector Field on the unit sphere S ^2 n +1 is the Reeb Vector Field of a natural Sasakian structure on S ^2 n +1. A contact metric manifold whose Reeb Vector Field ξ is a Harmonic Vector Field is called an H-contact manifold . Sasakian and K -contact manifolds, generalized ( k , μ )-spaces and contact metric three-manifolds with ξ strongly normal, are H -contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb Vector Field ξ for such special classes of H -contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb Vector Field of a compact K -contact (2 n +1)-manifold the obtained results for the Hopf Vector Fields to minimize the energy functional with mean curvature correction.
Sorin Dragomir - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Vector Fields
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary This chapter discusses fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent Vector Field on a Riemannian manifold, first and second variation formulae, and the Harmonic Vector Fields system. The study of the weak solutions to this system (existence and local properties) is missing from the present day mathematical literature. Various instances are investigated where Harmonic Vector Fields occur and to generalizations. Any unit Vector Field that is a Harmonic map is also a Harmonic Vector Field. The study of Harmonic map system is more appropriate on a Hermitian manifold and that results in Hermitian Harmonic maps to be useful in studying rigidity of complete Hermitian manifolds.
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Geometry of the Tangent Bundle
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary This chapter briefly reviews the basic facts in the geometry of the tangent bundle over a Riemannian manifold, such as nonlinear connections, the Dombrowski map, and the Sasaki metric. Tangent bundle also carries a natural almost complex structure compatible to Sasaki metric and such that tangent bundle and complex structure are almost Kahler manifold. The results offer the possibility of further development for the Harmonic Vector Field theory. A deep circle of ideas relates the geometry of the tangent bundle over a Riemannian manifold to the study of the global solutions to the homogeneous complex Monge-Ampere equation. It further examines the behavior of the total bending functional under conformational transformation. Conclusive and complete results on Harmonic Vector Fields on Riemannian tori are presented.
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Chapter Five – Harmonicity with Respect to g-Natural Metrics
Harmonic Vector Fields, 2020Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:Publisher Summary The Harmonic map equations for unit Vector Fields are discussed for the particular case of Reeb Fields in contact metric geometry. It is always a Harmonic Vector Field and it is a Harmonic map under appropriate conditions on the coefficients determining the fixed g-natural metric, allowing one to exhibit large families of Harmonic maps defined on a compact Riemannian manifold and having a target space with a highly nontrivial geometry. Vertical Harmonicity with respect to Riemannian g-natural metrics appears to be worth further investigation. In the theory of Harmonic maps, a fundamental question concerns the existence of Harmonic maps between two given Riemannian manifolds. It is therefore important to find examples of Harmonic maps into Riemannian manifolds whose sectional curvature is not necessarily non-positive.
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On the geometry of tangent hyperquadric bundles: CR and pseudoHarmonic Vector Fields
Annals of Global Analysis and Geometry, 2006Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T ( M ) over a semi-Riemannian manifold ( M , g ) and show that if the Reeb Vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a Harmonic Vector Field. As an other application, we study pseudoHarmonic Vector Fields on a compact strictly pseudoconvex CR manifold M , i.e. unit (with respect to the Webster metric associated with a fixed contact form on M ) Vector Fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S^1 → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C ( M )). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M .
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On the geometry of tangent hyperquadric bundles: CR and pseudoHarmonic Vector Fields
Annals of Global Analysis and Geometry, 2006Co-Authors: Sorin Dragomir, Domenico PerroneAbstract:We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb Vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a Harmonic Vector Field. As an other application, we study pseudoHarmonic Vector Fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) Vector Fields X e H(M) whose horizontal lift X↑ to the canonical circle bundle S1 → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X↑ project on a nonlinear system of subelliptic PDEs on M.
Yunbo He - One of the best experts on this subject based on the ideXlab platform.
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Triangular Model Registration Algorithm Through Differential Topological Singularity Points by Helmholtz-Hodge Decomposition
IEEE Access, 2019Co-Authors: Dongqing Wu, Zhengtao Xiao, Lanyu Zhang, Xin Chen, Hui Tang, Yun Chen, Yunbo HeAbstract:Iterative closest point algorithms suffer from non-convergence and local minima when dealing with cloud points with a different sampling density. Alternative global or semi-global registration algorithms may suffer from efficiency problem. This paper proposes a new registration algorithm through the differential topological singularity points (DTSP) based on the Helmholtz-Hodge decomposition (HHD), which is called DTSP-ICP method. The DTSP-ICP method contains two algorithms. First, the curvature gradient Fields on surfaces are decomposed by the HHD into three orthogonal parts: divergence-free Vector Field, curl-free Vector Field, and a Harmonic Vector Field, and then the DTSP algorithm is used to extract the differential topological singularity points in the curl-free Vector Field. Second, the ICP algorithm is utilized to register the singularity points into one aligned model. The singularity points represent the feature of the whole model, and the DTSP algorithm is designed to capture the nature of the differential topological structure of a mesh model. Through the singularity alignment, the DTSP-ICP method, therefore, possesses better performance in triangular model registration. The experimental results show that independent of sampling schemes, the proposed DTSP-ICP method can maintain convergence and robustness in cases where other alignment algorithms including the ICP alone are unstable. Moreover, this DTSP-ICP method can avoid the local errors of model registration based on Euclidean distance and overcome the computation insufficiencies observed in other global or semi-global registration publications. Finally, we demonstrate the significance of the DTSP-ICP algorithm's advantages on a variety of challenging models through result comparison with that of two other typical methods.
Dongqing Wu - One of the best experts on this subject based on the ideXlab platform.
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Triangular Model Registration Algorithm Through Differential Topological Singularity Points by Helmholtz-Hodge Decomposition
IEEE Access, 2019Co-Authors: Dongqing Wu, Zhengtao Xiao, Lanyu Zhang, Xin Chen, Hui Tang, Yun Chen, Yunbo HeAbstract:Iterative closest point algorithms suffer from non-convergence and local minima when dealing with cloud points with a different sampling density. Alternative global or semi-global registration algorithms may suffer from efficiency problem. This paper proposes a new registration algorithm through the differential topological singularity points (DTSP) based on the Helmholtz-Hodge decomposition (HHD), which is called DTSP-ICP method. The DTSP-ICP method contains two algorithms. First, the curvature gradient Fields on surfaces are decomposed by the HHD into three orthogonal parts: divergence-free Vector Field, curl-free Vector Field, and a Harmonic Vector Field, and then the DTSP algorithm is used to extract the differential topological singularity points in the curl-free Vector Field. Second, the ICP algorithm is utilized to register the singularity points into one aligned model. The singularity points represent the feature of the whole model, and the DTSP algorithm is designed to capture the nature of the differential topological structure of a mesh model. Through the singularity alignment, the DTSP-ICP method, therefore, possesses better performance in triangular model registration. The experimental results show that independent of sampling schemes, the proposed DTSP-ICP method can maintain convergence and robustness in cases where other alignment algorithms including the ICP alone are unstable. Moreover, this DTSP-ICP method can avoid the local errors of model registration based on Euclidean distance and overcome the computation insufficiencies observed in other global or semi-global registration publications. Finally, we demonstrate the significance of the DTSP-ICP algorithm's advantages on a variety of challenging models through result comparison with that of two other typical methods.
Yun Chen - One of the best experts on this subject based on the ideXlab platform.
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Triangular Model Registration Algorithm Through Differential Topological Singularity Points by Helmholtz-Hodge Decomposition
IEEE Access, 2019Co-Authors: Dongqing Wu, Zhengtao Xiao, Lanyu Zhang, Xin Chen, Hui Tang, Yun Chen, Yunbo HeAbstract:Iterative closest point algorithms suffer from non-convergence and local minima when dealing with cloud points with a different sampling density. Alternative global or semi-global registration algorithms may suffer from efficiency problem. This paper proposes a new registration algorithm through the differential topological singularity points (DTSP) based on the Helmholtz-Hodge decomposition (HHD), which is called DTSP-ICP method. The DTSP-ICP method contains two algorithms. First, the curvature gradient Fields on surfaces are decomposed by the HHD into three orthogonal parts: divergence-free Vector Field, curl-free Vector Field, and a Harmonic Vector Field, and then the DTSP algorithm is used to extract the differential topological singularity points in the curl-free Vector Field. Second, the ICP algorithm is utilized to register the singularity points into one aligned model. The singularity points represent the feature of the whole model, and the DTSP algorithm is designed to capture the nature of the differential topological structure of a mesh model. Through the singularity alignment, the DTSP-ICP method, therefore, possesses better performance in triangular model registration. The experimental results show that independent of sampling schemes, the proposed DTSP-ICP method can maintain convergence and robustness in cases where other alignment algorithms including the ICP alone are unstable. Moreover, this DTSP-ICP method can avoid the local errors of model registration based on Euclidean distance and overcome the computation insufficiencies observed in other global or semi-global registration publications. Finally, we demonstrate the significance of the DTSP-ICP algorithm's advantages on a variety of challenging models through result comparison with that of two other typical methods.
