# Connections on Complex Finsler Manifold

Connections on Complex Finsler Manifold We introduce Finsler metric on complex manifold and discuss connections induced by this metric. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Connections on Complex Finsler Manifold

, Volume 19 (3) – Mar 3, 2017

## Connections on Complex Finsler Manifold

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 431–436 Rong-mu Yan Department of Mathematics, Xiamen University, Xiamen 361005, China (E-mail: yanrm@jingxian.xmu.edu.cn) Abstract We introduce Finsler metric on complex manifold and discuss connections induced by this metric. Keywords Complex Minkowski functional, complex Finsler manifold, Cartan scalar, Berwald scalar 2000 MR Subject Classiﬁcation 53B40, 32H02 Ever since Finsler’s pioneering work in 1918, the topic of Finsler geometry has been considered [4,6,7] by many geometers . Many applications to biology and physics have also been made (see [1]). In this paper, we introduce Finsler metric on complex manifold and discuss connections under this metric. We also compare these connections with (1,0)-compatible connection on [1,4,6,7] Hermitian manifold. In this article, we introduce Finsler metric on complex manifold and discuss connections induced by this metric. We would also compare these connections with (1,0)-compatible connection on Hermitian manifold. 1 Basic Deﬁnition Deﬁnition 1. Let V be an n-dimension complex linear space. A complex Minkowski func- tional F on V is a function F : V → [0,∞) which has the following properties (i) F is module homogeneous, i.e. F (λy)= |λ|F (y), ∀λ ∈ C, ∀y ∈ V , (ii) F is C on V −{0}, (iii) ∀y ∈ V −{0}, the fundamental form g is an inner product, where g (v ,v )= F (y + z v + z v ) . (1) y 1 2 1 1 2 2 z =z =0 1 2 ∂z ∂z 1 2 2 2 ∂ [F ] For a basis {e } for V and y = y e ,let g (y)= g (e ,e ), we have g (y)= ,so(g ) i i y i j j ij ij ij ∂y ∂y is a Hermitian matrix. And ∂ ∂ 2 2 g (y, v)= [F (y + zv)] ,g (v, y)= [F (y + zv)] , (2) y z=0 y z=0 ∂z ∂z g (y, y)= F (y). (3) From this and the deﬁnition of F,wehave g (λy)= g (y), ∀λ ∈ C\{0}. (4) ij ij Manuscript received December 12, 2000....

/lp/springer_journal/connections-on-complex-finsler-manifold-WAGfJqGUu4
Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0118-y
Publisher site
See Article on Publisher Site

### Abstract

We introduce Finsler metric on complex manifold and discuss connections induced by this metric.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

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