Volume 55, Issue 2
On Doubly Twisted Product of Complex Finsler Manifolds

J. Math. Study, 55 (2022), pp. 158-179.

Published online: 2022-04

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• Abstract

Let $(M_1,F_1)$ and $(M_2,F_2)$ be two strongly pseudoconvex complex Finsler manifolds. The doubly twisted product (abbreviated as DTP) complex Finsler manifold $(M_1\times_{(\lambda_1,\lambda_2)}M_2,F)$ is the product manifold $M_1\times M_2$ endowed with the twisted product complex Finsler metric $F^2=\lambda_1^2F_1^2+\lambda_2^2F_2^2$, where $\lambda_1$ and $\lambda_2$ are positive smooth functions on $M_1\times M_2$. In this paper, the relationships between the geometric objects (e.g. complex Finsler connections, holomorphic and Ricci scalar curvatures, and real geodesic) of a DTP-complex Finsler manifold and its components are derived. The necessary and sufficient conditions under which the DTP-complex Finsler manifold is a Kähler Finsler (respctively weakly Kähler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained. By means of these, we provide a possible way to construct a weakly complex Berwald manifold, and then give a characterization for a complex Landsberg metric that is not a Berwald metric.

• Keywords

Doubly twisted products, complex Finsler metric, holomorphic curvature, geodesic.

53C60, 53C40

xiaow9704@126.com (Wei Xiao)

heyong7905@126.com (Yong He)

luxy8516@126.com (Xiaoying Lu)

dxxw523@126.com (Xiangxiang Deng)

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@Article{JMS-55-158, author = {Wei and Xiao and xiaow9704@126.com and 23076 and School of Mathematical Sciences, XinJiang Normal University, Urumqi 830017, China and Wei Xiao and Yong and He and heyong7905@126.com and 23077 and School of Mathematical Sciences, XinJiang Normal University, Urumqi 830017, China and Yong He and Xiaoying and Lu and luxy8516@126.com and 23078 and Urumqi Campus of Army Academy of Border and Coastal Defence, Urumqi 830002, China and Xiaoying Lu and Xiangxiang and Deng and dxxw523@126.com and 23079 and School of Mathematical Sciences, XinJiang Normal University, Urumqi 830017, China and Xiangxiang Deng}, title = {On Doubly Twisted Product of Complex Finsler Manifolds}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {2}, pages = {158--179}, abstract = {

Let $(M_1,F_1)$ and $(M_2,F_2)$ be two strongly pseudoconvex complex Finsler manifolds. The doubly twisted product (abbreviated as DTP) complex Finsler manifold $(M_1\times_{(\lambda_1,\lambda_2)}M_2,F)$ is the product manifold $M_1\times M_2$ endowed with the twisted product complex Finsler metric $F^2=\lambda_1^2F_1^2+\lambda_2^2F_2^2$, where $\lambda_1$ and $\lambda_2$ are positive smooth functions on $M_1\times M_2$. In this paper, the relationships between the geometric objects (e.g. complex Finsler connections, holomorphic and Ricci scalar curvatures, and real geodesic) of a DTP-complex Finsler manifold and its components are derived. The necessary and sufficient conditions under which the DTP-complex Finsler manifold is a Kähler Finsler (respctively weakly Kähler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained. By means of these, we provide a possible way to construct a weakly complex Berwald manifold, and then give a characterization for a complex Landsberg metric that is not a Berwald metric.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n2.22.04}, url = {http://global-sci.org/intro/article_detail/jms/20493.html} }
TY - JOUR T1 - On Doubly Twisted Product of Complex Finsler Manifolds AU - Xiao , Wei AU - He , Yong AU - Lu , Xiaoying AU - Deng , Xiangxiang JO - Journal of Mathematical Study VL - 2 SP - 158 EP - 179 PY - 2022 DA - 2022/04 SN - 55 DO - http://doi.org/10.4208/jms.v55n2.22.04 UR - https://global-sci.org/intro/article_detail/jms/20493.html KW - Doubly twisted products, complex Finsler metric, holomorphic curvature, geodesic. AB -

Let $(M_1,F_1)$ and $(M_2,F_2)$ be two strongly pseudoconvex complex Finsler manifolds. The doubly twisted product (abbreviated as DTP) complex Finsler manifold $(M_1\times_{(\lambda_1,\lambda_2)}M_2,F)$ is the product manifold $M_1\times M_2$ endowed with the twisted product complex Finsler metric $F^2=\lambda_1^2F_1^2+\lambda_2^2F_2^2$, where $\lambda_1$ and $\lambda_2$ are positive smooth functions on $M_1\times M_2$. In this paper, the relationships between the geometric objects (e.g. complex Finsler connections, holomorphic and Ricci scalar curvatures, and real geodesic) of a DTP-complex Finsler manifold and its components are derived. The necessary and sufficient conditions under which the DTP-complex Finsler manifold is a Kähler Finsler (respctively weakly Kähler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg) manifold are obtained. By means of these, we provide a possible way to construct a weakly complex Berwald manifold, and then give a characterization for a complex Landsberg metric that is not a Berwald metric.

Wei Xiao, Yong He, Xiaoying Lu & Xiangxiang Deng. (2022). On Doubly Twisted Product of Complex Finsler Manifolds. Journal of Mathematical Study. 55 (2). 158-179. doi:10.4208/jms.v55n2.22.04
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