关于半欧氏空间之间调和同态的一些结论
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摘要
本文研究半欧氏空间R~mR_r→R~n_3之间的调和态射。通过推广
［12］和［14］中构造欧氏空间之间调和态射的方法，我们得出半欧
氏空问之间两个相应的定理（如下），并给出证明。同时，我们
也给出半欧氏空间之间二次调和态射的一些有趣例子。
定理2．3设是齐二次多项式调和态射，φ的完全
提升φ:定义为，那
么φ是调和态射。这里 R~M_r×R~M——r具有乘积度量形式
定理2.4设是两个调和态射，它
们的直和φ1_φ_2定义为，乘积流形
具有积度量，那么φ1_ φ_2是调和态射。
In this thesis , we study harmonic morphisms betweem semi-Euclidean
spaces R~ ? R~. We prove following two theorems which give two ways to
construct harmonic morphisms between semi-Euclidean spaces, thus gen-
eralize the corresponding results in [12] and [14]. We also give some inter-
esting examples of quadratic harmonic morphims between semi-Euclidean
spaces
Theorem 2.3
be a harmonic morphism defined by
homogeneous polynomial 01 degree ~ , tnen tne corn p.Lere UIL 01 9, uenneu
is a harmonic
morphism, where R~ x R~ is given the product metric having the form
Theorem 2.4
be two harmonic
morDhisrns. then the direct sum
is a harmonic mor
phism,where the product manifold MJ~ x is provided with the product
metric
［12］和［14］中构造欧氏空间之间调和态射的方法，我们得出半欧
氏空问之间两个相应的定理（如下），并给出证明。同时，我们
也给出半欧氏空间之间二次调和态射的一些有趣例子。
定理2．3设是齐二次多项式调和态射，φ的完全
提升φ:定义为，那
么φ是调和态射。这里 R~M_r×R~M——r具有乘积度量形式
定理2.4设是两个调和态射，它
们的直和φ1_φ_2定义为，乘积流形
具有积度量，那么φ1_ φ_2是调和态射。
In this thesis , we study harmonic morphisms betweem semi-Euclidean
spaces R~ ? R~. We prove following two theorems which give two ways to
construct harmonic morphisms between semi-Euclidean spaces, thus gen-
eralize the corresponding results in [12] and [14]. We also give some inter-
esting examples of quadratic harmonic morphims between semi-Euclidean
spaces
Theorem 2.3
be a harmonic morphism defined by
homogeneous polynomial 01 degree ~ , tnen tne corn p.Lere UIL 01 9, uenneu
is a harmonic
morphism, where R~ x R~ is given the product metric having the form
Theorem 2.4
be two harmonic
morDhisrns. then the direct sum
is a harmonic mor
phism,where the product manifold MJ~ x is provided with the product
metric
引文
[1] R Aobu, P Baird and J Borssard, Polynomes conformes et morphismes harmomques, preprint, University of Brest (1998)
[2] P Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Res Notes in Math , Vol 87, Pitman, London, 1983
[3] P Baird,Harmonic morphisms and circle 3-and 4-manifolds,Am Inst founer Grenoble 40(1990) ,177-212
[4] J Eells and L Lemaire, A report on harmonic maps, Bull London Math Soc 10 (1978) , 1-68
[5] J Eells and L Lemaire, Selected topics in harmonic maps, CMBS Regional Conference Series in Mathematics 50, Amer Math Soc 1983
[6] J Eells and L Lemaire, Another report on harmonic maps, Bull London Math Soc 20 (1998) , 385-524
[7] J Eells and PYiu, Polynomial harmonic morphisms between spheres, Proc Amer Math Soc 123 (1995) , 2921-2925
[8] B Fuglede, Harmonic morphism between Riemannian manifold, Ann Inst Fourier (Grenoble) 28 (1978) , 107-144
[9] B Puglede, Harmonic morphism between semi-Riemanman manifolds, Ann Acad Sci Fennicae 21 (1996) ,31-50
[10] T Ishihara, A mapping of Riemanman manifolds which preserves harmonic function,J Math Kyoto Univ 19 (1979) , 215-229
[11] B O'Neill,Semi-Riemannian Geometry with apphcations to relativity-Academic Press, New York, 1983
[12] Y L Ou, Complete lifts of maps and harmonic morphisms between Euclidean space, Beitrage Algebra Geom 1995
[13] Y L On, and J C Wood, On the classification of harmonic morphisms between Euclidean spaces, Algebra, Groups and Geometries 13 (1996) , 41-53
[14] Y L Ou, Quadratic harmonic morphisms and O-systems, Ann Inst Fourier (Grenoble) 47 (1997) , 687-713
[15] V K Parmar, Harmonic morphisms between semi-Riemannian manifolds,Ph D Thesis, University of Leeds, England, (1991)
[16] Martin Sevensson, Polynomial harmonic morphism,-
[17] Z Tang, Harmonic polynomial morphisms between Euclidean space, Preprint, ICTP Trieste (1996)
[2] P Baird, Harmonic maps with symmetry, harmonic morphisms and deformations of metrics, Res Notes in Math , Vol 87, Pitman, London, 1983
[3] P Baird,Harmonic morphisms and circle 3-and 4-manifolds,Am Inst founer Grenoble 40(1990) ,177-212
[4] J Eells and L Lemaire, A report on harmonic maps, Bull London Math Soc 10 (1978) , 1-68
[5] J Eells and L Lemaire, Selected topics in harmonic maps, CMBS Regional Conference Series in Mathematics 50, Amer Math Soc 1983
[6] J Eells and L Lemaire, Another report on harmonic maps, Bull London Math Soc 20 (1998) , 385-524
[7] J Eells and PYiu, Polynomial harmonic morphisms between spheres, Proc Amer Math Soc 123 (1995) , 2921-2925
[8] B Fuglede, Harmonic morphism between Riemannian manifold, Ann Inst Fourier (Grenoble) 28 (1978) , 107-144
[9] B Puglede, Harmonic morphism between semi-Riemanman manifolds, Ann Acad Sci Fennicae 21 (1996) ,31-50
[10] T Ishihara, A mapping of Riemanman manifolds which preserves harmonic function,J Math Kyoto Univ 19 (1979) , 215-229
[11] B O'Neill,Semi-Riemannian Geometry with apphcations to relativity-Academic Press, New York, 1983
[12] Y L Ou, Complete lifts of maps and harmonic morphisms between Euclidean space, Beitrage Algebra Geom 1995
[13] Y L On, and J C Wood, On the classification of harmonic morphisms between Euclidean spaces, Algebra, Groups and Geometries 13 (1996) , 41-53
[14] Y L Ou, Quadratic harmonic morphisms and O-systems, Ann Inst Fourier (Grenoble) 47 (1997) , 687-713
[15] V K Parmar, Harmonic morphisms between semi-Riemannian manifolds,Ph D Thesis, University of Leeds, England, (1991)
[16] Martin Sevensson, Polynomial harmonic morphism,-
[17] Z Tang, Harmonic polynomial morphisms between Euclidean space, Preprint, ICTP Trieste (1996)