Berndt J., Tricerri F., Vanhecke L. — Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces :: Электронная библиотека попечительского совета мехмата МГУ

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Berndt J., Tricerri F., Vanhecke L. — Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

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Название: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

Авторы: Berndt J., Tricerri F., Vanhecke L.

Аннотация:

Generalized Heisenberg groups, or H-type groups, introduced by A. Kaplan, and Damek-Ricci harmonic spaces are particularly nice Lie groups with a vast spectrum of properties and applications. These harmonic spaces are homogeneous Hadamard manifolds containing the H-type groups as horospheres. These notes contain a thorough study of their Riemannian geometry by means of a detailed treatment of their Jacobi vector fields and Jacobi operators. Some problems are included and will hopefully stimulate further research on these spaces. The book is written for students and researchers, assuming only basic knowledge of Riemannian geometry, and it contains a brief survey of the background material needed to follow the entire treatment.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 125

Добавлена в каталог: 11.04.2010

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Предметный указатель

$J^{2}$ -condition      25 33 39 86 104
$J_{z}$ -operator      22
-harmonic spaces      113
-operator      33
-symmetric spaces      4
$T_{\gamma}$ -tensor      45
$\bar{\nabla}_{\partial}$ -operator      51
$\eta$ -umbilical hypersurface      7
$\mathfrak{B}$ -space      14 18 44 107
$\mathfrak{B}$ -space of dimension three      14-15
$\mathfrak{B}$ -space of dimension two      14
$\mathfrak{C}$ -space      14 18 19 42 50 105 111
$\mathfrak{C}$ -space, of dimension three      14
$\mathfrak{C}$ -space, of dimension two      14
$\mathfrak{C}_{0}$ -space      16 18 19 50 111
$\mathfrak{C}_{0}$ -space, of dimension three      17
$\mathfrak{SC}$ -space      18 50 70 111
$\mathfrak{SC}$ -space of dimension $\leq 3$       19
$\mathfrak{TB}$ - space      18
$\mathfrak{TC}$ -space      17 19 50 74 111
$\mathfrak{TC}$ -space of dimension $\leq 3$       18
$\varphi$ -symmetric space      7-8
Alekseevskii space      96
Analyticity of $\mathbb{C}$ -spaces      16
Analyticity of D’Atri spaces      13
Asymptotically harmonic      12
Basic Jacobi vector field      16
Bergman metric      91
Cayley transform      91
Classical Heisenberg algebra      24 79
Clifford algebra      22
Clifford module      22-23
Commutative space      10 12 13 15 17 18 19 35 74 111 112
Commutative space, of dimension $\leq 5$       10
Conjugate points on generalized Heisenberg groups      66
Constant mean curvature      12 110 113
Constant scalar curvature of $\mathfrak{C}$ -spaces      16
Constant scalar curvature of D’Atri spaces      13
Constant scalar curvature of geodesic spheres      12
Curvature tensor of Damek — Ricci spaces      84-85
Curvature tensor of generalized Heisenberg groups      28
Curvature-adapted geodesic spheres      16
Curvature-homogeneous manifold      112
Damek — Ricci space      79
Distance function on Damek — Ricci spaces      108
D’Atri space      13 17 18 19 36 50 74 77 111 112 114
D’Atri space of dimension four      14
D’Atri space of dimension three      13
Euclidean Laplacian      11
Extrinsic sphere      87
First mean value operator      12
Fundamental conjecture about harmonic manifolds      12
Generalized Heisenberg algebra      22
Generalized Heisenberg group      22
Geodesic flow      16
Geodesic spheres      6 7 8 9 12 13 16 18 69 112 113
Geodesic spray      16
Geodesics on Damek — Ricci spaces      93
Geodesics on generalized Heisenberg groups      31
Global coordinates on Damek — Ricci spaces      82
Global coordinates on generalized Heisenberg groups      26
Hadamard manifold      85 113 114
Harmonic function      11 12
Harmonic space      11 13 36 110 112 113 114
Harmonic space of dimension $\leq 4$       12
Harmonicity of Damek — Ricci spaces      108
Heisenberg group of dimension five      7
Heisenberg group of dimension three      6
Homogeneous structure      5 51
Horosphere      6 9 88
Horosphere foliation      88
Integrability of subbundles on Damek — Ricci spaces      87
Integrability of subbundles on generalized Heisenberg groups      32
Irreducibility of Damek — Ricci spaces      111
Irreducibility of generalized Heisenberg groups      32
Isometry group of Damek — Ricci spaces      94
Isometry group of generalized Heisenberg groups      34
Isoparametric function      110
Isoparametric geodesic spheres      113

Isoparametric hypersurface      88 110
Iwasawa decomposition      25 79
Jacobi equation      51
Jacobi operator      14 19
Jacobi operator on Damek — Ricci spaces      85 96
Jacobi operator on generalized Heisenberg groups      29 36
Jacobi operators of higher order      17
Jacobi vector field      16
Jacobi vector field on generalized Heisenberg groups      52
K $\ddt{a}$ hler structure on Damek — Ricci spaces      96
K $\ddt{a}$ hler structure on generalized Heisenberg groups      34
Killing tensor      14 16 29 112
Killing-transversally symmetric space      8
Laplace equation      11
Laplace operator on Damek — Ricci spaces      108
Ledger tensor      15
Levi — Civita connection on Damek — Ricci spaces      84
Levi — Civita connection on generalized Heisenberg groups      28
Lie exponential map for Damek — Ricci spaces      80
Lie exponential map for generalized Heisenberg groups      26
Liouville surface      14
Mean curvature of geodesic spheres      13
Metric tensor in normal coordiantes on generalized Heisenberg groups      74
Natural torsion-free connection      4-5
Naturally reductive Riemannian homogeneous space      4 8 9 10 11 13 14 17 18 19 35 74 111
Naturally reductive Riemannian homogeneous space of dimension five      7
Naturally reductive Riemannian homogeneous space of dimension four      6
Naturally reductive Riemannian homogeneous space of dimension three      6
Naturally reductive Riemannian homogeneous space of dimension two      5
Nearly K $\ddot{a}$ hler structure      95
Nearly K $\ddot{a}$ hler structure on Damek — Ricci spaces      95
Osserman conjecture      19
Osserman space      19
Osserman space, globally      19 111 112 113
Osserman space, pointwise      20 42
Principal curvature spaces of geodesic spheres      16
Principal curvatures of geodesic spheres      18 113
Principal curvatures of geodesic spheres on generalized Heisenberg groups      69
Probabilistic commutative space      10 12 13 111 112
Probabilistic commutative space of dimension three      11
Quaternionic K $\ddot{a}$ hler structure on Damek — Ricci spaces      96
Quotients of finite volume of Damek — Ricci spaces      112
Ray symmetric space      9
Reductive Lie algebra      4
Ricci flat harmonic spaces      113
Ricci tensor of Damek — Ricci spaces      85
Ricci tensor of generalized Heisenberg groups      28
Ricci tensor of geodesic spheres      113
Riemannian foliation      87
Riemannian g.o. space      8 10 13 15 17 18 35 50 74 111
Riemannian g.o. space of dimension $\leq 5$       8
Riemannian g.o. space of dimension six      8
Riemannian submersion      87
Sasaki metric      16
Sasakian space form      18 19
Scalar curvature of Damek — Ricci spaces      85
Scalar curvature of generalized Heisenberg groups      28
Scalar curvature of geodesic spheres      74
Second mean value operator      10
Sectional curvature of Damek — Ricci spaces      85 104-105
Sectional curvature of generalized Heisenberg groups      29
Self-dual Einstein space of dimension four      20
Semi-symmetric manifold      112
Shape operator of geodesic spheres      16 18
Solvable Lie group      80
Special curvature tensor      17
Spherical submanifold      87
Symmetric Damek-Ricci spaces      86 105 107 111 112 113
Symmetric-like Riemannian manifolds      4
Transnormal function      110
Tubes      6 9
Two-step nilpotent Lie algebra      23
Two-step nilpotent Lie group      8 26
Volume density function      11
Volume-preserving geodesic symmetry      13
Warped product      14
Weakly symmetric space      9 10 18 19 36 111
Weakly symmetric space of dimension four      9
Weakly symmetric space of dimension three      9

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