scholarly journals Some notes on metallic Kähler manifolds

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1963-1975
Author(s):  
Aydin Gezer ◽  
Fatih Topcuoglu ◽  
De Chand
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Hermitian Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemannian Curvature ◽  
Tensor Form ◽  
Riemannian Curvature Tensor ◽  
Zero Scalar Curvature ◽  
Conformally Recurrent

The present paper deals with metallic K?hler manifolds. Firstly, we define a tensor H which can be written in terms of the (0,4)-Riemannian curvature tensor and the fundamental 2-form of a metallic K?hler manifold and study its properties and some hybrid tensors. Secondly, weobtain the conditions under which a metallic Hermitian manifold is conformal to a metallic K?hler manifold. Thirdly, we prove that the conformal recurrency of a metallic K?hler manifold implies its recurrency and also obtain the Riemannian curvature tensor form of a conformally recurrent metallic K?hler manifold with non-zero scalar curvature. Finally, we present a result related to the notion of Z recurrent form on a metallic K?hler manifold.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

scholarly journals On some compact almost Kähler locally symmetric space

1998 ◽  
Vol 21 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Takashi Oguro
Keyword(s):  
Symmetric Space ◽  
Scalar Curvature ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Locally Symmetric Space ◽  
Kahler Manifold ◽  
Almost Kähler

In the framework of studying the integrability of almost Kähler manifolds, we prove that if a compact almost Kähler locally symmetric spaceMis a weakly ,∗-Einstein vnanifold with non-negative ,∗-scalar curvature, thenMis a Kähler manifold.

scholarly journals Blowing up and desingularizing constant scalar curvature Kähler manifolds

2006 ◽  
Vol 196 (2) ◽  
pp. 179-228 ◽  
Author(s):  
Claudio Arezzo ◽  
Frank Pacard
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Blowing Up

On the geometry of conharmonic curvature tensor for nearly Kähler manifolds

2011 ◽  
Vol 177 (5) ◽  
pp. 675-683
Author(s):  
V. F. Kirichenko ◽  
A. A. Shihab
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Conharmonic Curvature Tensor ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

scholarly journals K-stability of constant scalar curvature Kähler manifolds

2009 ◽  
Vol 221 (4) ◽  
pp. 1397-1408 ◽  
Author(s):  
Jacopo Stoppa
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds

scholarly journals Nearly Kähler manifolds with vanishing Tricerri–Vanhecke Bochner curvature tensor

2009 ◽  
Vol 27 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Y. Euh ◽  
J.H. Park ◽  
K. Sekigawa
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

scholarly journals A (CHR)3-flat trans-Sasakian manifold

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Hermitian Manifolds ◽  
Riemannian Curvature Tensor ◽  
Contact Riemannian Manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

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