Isotropic and Kähler Immersions

1965 ◽  
Vol 17 ◽  
pp. 907-915 ◽  
Author(s):  
Barrett O'Neill
Keyword(s):  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Isometric Immersion ◽  
Normal Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Real Constant ◽  
Cartesian Products ◽  
The Real ◽  
Holomorphic Curvature

Let Md and be Riemannian manifolds. We shall say that an isometric immersion ϕ: Md —> is isotropic provided that all its normal curvature vectors have the same length. The class of such immersions is closed under compositions and Cartesian products. Umbilic immersions (e.g. Sd ⊂ Rd+1) are isotropic, but the converse does not hold. If M and are Kähler manifolds of constant holomorphic curvature, then any Kähler immersion of M in is automatically isotropic (Lemma 6). We shall find the smallest co-dimension for which there exist non-trivial immersions of this type, and obtain similar results in the real constant-curvature case.

scholarly journals PARTIAL CLASSIFICATION RESULTS FOR POSITIVE QUATERNION KÄHLER MANIFOLDS

2012 ◽  
Vol 23 (02) ◽  
pp. 1250038
Author(s):  
MANUEL AMANN
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Positive Scalar Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
The Real ◽  
Positive Scalar ◽  
Partial Classification ◽  
Almost All

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for the real Grassmannian [Formula: see text] in almost all dimensions.

ON HARMONIC AND PSEUDOHARMONIC MAPS FROM PSEUDO-HERMITIAN MANIFOLDS

2017 ◽  
Vol 234 ◽  
pp. 170-210 ◽  
Author(s):  
TIAN CHONG ◽  
YUXIN DONG ◽  
YIBIN REN ◽  
GUILIN YANG
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Rigidity Results ◽  
Hermitian Manifolds

In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from pseudo-Hermitian manifolds into Riemannian manifolds or Kähler manifolds. Some foliated results, pluriharmonicity and Siu–Sampson type results are established for both harmonic maps and pseudoharmonic maps.

scholarly journals Metallic Kähler and nearly metallic Kähler manifolds

2021 ◽  
pp. 2150146
Author(s):  
Sibel Turanli ◽  
Aydin Gezer ◽  
Hasan Cakicioglu
Keyword(s):  
Riemannian Manifolds ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Linear Connections ◽  
Kähler Structures

In this paper, we construct metallic Kähler and nearly metallic Kähler structures on Riemannian manifolds. For such manifolds with these structures, we study curvature properties. Also, we describe linear connections on the manifold which preserve the associated fundamental 2-form and satisfy some additional conditions and present some results concerning them.

A SCHWARZ LEMMA FOR -HARMONIC MAPS AND THEIR APPLICATIONS

2017 ◽  
Vol 96 (3) ◽  
pp. 504-512 ◽  
Author(s):  
QUN CHEN ◽  
GUANGWEN ZHAO
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Schwarz Lemma ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Holomorphic Maps ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Almost Kähler

We establish a Schwarz lemma for $V$-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.

scholarly journals On the geometry of affine Kähler immersions

1990 ◽  
Vol 120 ◽  
pp. 205-222 ◽  
Author(s):  
Katsumi Nomizu ◽  
Ulrich Pinkall ◽  
Fabio Podestà
Keyword(s):  
Isometric Immersion ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Complex Manifolds ◽  
Affine Immersion ◽  
Holomorphic Affine ◽  
Affine Immersions ◽  
Holomorphic Affine Immersion

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].

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