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  浙江大学学报(理学版)  2017, Vol. 44 Issue (2): 154-160  DOI:10.3785/j.issn.1008-9497.2017.02.006
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翁桂英. 复Finsler度量射影等价[J]. 浙江大学学报(理学版), 2017, 44(2): 154-160. DOI: 10.3785/j.issn.1008-9497.2017.02.006.
[复制中文]
WENG Guiying. Projectively equivalent complex Finsler metrics[J]. Journal of Zhejiang University(Science Edition), 2017, 44(2): 154-160. DOI: 10.3785/j.issn.1008-9497.2017.02.006.
[复制英文]

作者简介

翁桂英 (1983-), ORCID:http://orcid.org/0000-0002-3469-1466, 女, 硕士, 讲师, 主要从事多复变数和复Finsler几何研究, E-mail:yeuwgy@163.com

文章历史

收稿日期:2016-01-28
复Finsler度量射影等价
翁桂英     
仰恩大学 数学系, 福建 泉州 362014
摘要: 主要研究复流形上复Finsler度量射影等价及仿射等价的若干充要条件,讨论了复Finsler流形上的测地线及2种平行移动,从而得到复Finsler度量仿射等价的另一充要条件,并将其应用于乘积复Finsler流形中.
关键词: 复Finsler度量    测地线    射影等价    仿射等价    乘积复Finsler度量    
Projectively equivalent complex Finsler metrics
WENG Guiying     
Department of Mathematics, Yangen University, Quanzhou 362014, Fujian Province, China
Abstract: We study the necessary and sufficient conditions in which two complex Finsler metrics F and $ \tilde F $ on a manifold are projectively equivalent. We discuss the geodesics and two kinds of parallel translations on a complex Finsler manifold. Then, another necessary and sufficient condition in which two complex Finsler metrics are affinely equivalent is obtained, and is applied to product complex Finsler manifold.
Key words: complex Finsler metrics    geodesics    projectively equivalent    affinely equivalent    product complex Finsler manifold    

流形上Finsler度量的射影等价性是Finsler几何的一个重要课题,文献[1]在实Finsler度量下研究了一般射影等价及仿射等价成立的充要条件.复Finsler度量的射影等价这一概念由YAN[2]和ALDEA等[3-6]于2012年引入并进行研究.本文将对复Finsler度量的射影等价、仿射等价性及复Finsler流形上的2种平行移动进行研究并给出其应用例子.

1 预备知识

首先, 简单介绍本文所需的一些记号,更多细节参见文献[7].

假设M是一个n维的复流形, 复化切丛TM=T1, 0MT0, 1M.其中T1, 0M为全纯切丛, (zα, vα)α=1, n表示T1, 0M上的局部坐标, 其坐标变换公式为zBα=zBα(zA), $ v_{_B}^{^\alpha } = \frac{{\partial z_{_B}^{^\alpha }}}{{\partial z_{_A}^{^\beta }}}v_{_A}^{^\beta } $.而T0, 1M=T1, 0M.记$ {\tilde M} $=T1, 0M\{0}, 其复化切丛$ {{T}_{\mathbb{C}}}\tilde{M}={{T}^{1, 0}}\tilde{M}\oplus {{T}^{0, 1}}\tilde{M} $自然的局部标架为$ {\left\{ {\frac{\partial }{{\partial {z^\alpha }}}, \frac{\partial }{{\partial {v^\alpha }}}} \right\}_{\alpha = \overline {1, n} }} $, 简记为${\{ {\partial _\alpha }, {{\dot \partial }_\alpha }\} _{\alpha = \overline {1, n} }} $, 其坐标变换的Jacobian矩阵见文献[7].

设垂直丛$ V\left( {\tilde M} \right) = {\rm{ker }}{{\rm{\pi }}_*} \subset {T^{1, 0}}\tilde M $, 其局部标架为$ {\{ {{\dot \partial }_\alpha }\} _{\alpha = \overline {1, n} }} $.任一复化非线性联络 (简记为 (c.n.c.)) 决定了一水平丛, 即$ {T^{1, 0}}\tilde M = H\left( {\tilde M} \right) \oplus V\left( {\tilde M} \right) $也决定了一适当的水平标架$ {\{ {\delta _\alpha } = {\partial _\alpha }-\Gamma _{_{;\alpha }}^{^\beta }{{\dot \partial }_\beta }\} _{\alpha = \overline {1, n} }} $, 此处Γ; αβ为 (c.n.c.) 的系数.

定义1[7] 复流形M上的连续非负函数F:T1, 0MR+若满足下列条件, 则称其为复Finsler度量:

1) G:=F2$ {\tilde M} $上光滑;

2) 任意v$ {\tilde M} $, F(v)>0;

3) 任意vT1, 0M, ξ∈ℂ, F(ξv)=|ξ|F(v).

赋有复Finsler度量的复流形称为复Finsler流形,简记为 (M, F).

本文用分号区分对底流形坐标及纤维坐标zα, vα的求导, 如G=G(z, v) 为T1, 0M上的光滑函数, $ {G_{\alpha \bar \beta }}: = \frac{{{\partial ^2}G}}{{\partial {v^\alpha }\partial \overline {{v^\beta }} }} $, $ {G_{\alpha ;\bar \beta }}: = \frac{{{\partial ^2}G}}{{\partial {v^\alpha }\partial \overline {{z^\beta }} }} $; 又记G|α:=δαG=G; α; αβGβ. $ {\tilde M} $上函数P(z, v) 若满足对任意的ξ∈ℂ, 有$ P\left( {z, \xi v} \right) = {\xi ^n}\overline {{\xi ^m}} P\left( {z, v} \right) $, 则称P(z, v) 为 (n, m) 齐次.

定义2[7] 若${\tilde M} $上的Levi矩阵 (Gαβ) 正定, 则称复Fisler度量F为强拟凸的.

下文若不特别说明, 复Finsler度量总表示强拟凸的.

定理1[7] 复Finsler流形 (M, F) 上, 〈, 〉为由F诱导的垂直丛V上的Hermitian结构, 则存在唯一的复垂直联络$ D:\chi \left( V \right)\to \chi (T_{\mathbb{C}}^{*}\tilde{M}\otimes V) $, 对任意的$X\in {{T}^{1, 0}}\tilde{M} $, V, Wχ(V), 有

$ X\left\langle {V,W} \right\rangle = \left\langle {{\nabla _X}V,W} \right\rangle + \left\langle {V,{\nabla _{\bar X}}W} \right\rangle , $ (1)

且称D为 (M, F) 上的Chern-Finsler (c.n.c.) 联络.

在局部坐标下, Chern-Finsler (c.n.c.) 联络系数为

$ \begin{array}{*{20}{c}} {\Gamma _{;\beta }^\alpha {G^{\bar \nu \alpha }}{G_{\bar \nu ;\beta }} = \Gamma _{\tau ;\beta }^\alpha {v^\tau },\;\;\;\;\Gamma _{\tau ;\beta }^\alpha = {G^{\bar \nu \alpha }}\left( {{\delta _\beta }{G_{\tau \bar \nu }}} \right),}\\ {\Gamma _{\tau \beta }^\alpha = {G^{\bar \nu \alpha }}{G_{\tau \bar \nu \beta }},} \end{array} $ (2)

其中 (Gvα) 为 (Gαβ) 的逆矩阵, 且∇δαδββ; ανδν, $ {{\Delta }_{{{{\dot{\partial }}}_{\alpha }}}}{{{\dot{\partial }}}_{\beta }}=\Gamma _{_{\beta \alpha }}^{^{\nu }}{{{\dot{\partial }}}_{\nu }} $.测地系数$ {{G}^{\alpha }}:=\frac{1}{2}\text{ }\!\!\Gamma\!\!\text{ }_{;\tau }^{^{\alpha }}{{v}^{\tau }} $为 (2, 0) 齐次, 即对于任意的ξ∈ℂ, 有Gα(z, ξv)=ξ2Gα(z, v), 因此vτ(Gα)τ=2Gα; ταvτ.

根据文献[7], 复Finsler流形是强Kähler当且仅当Γμ; ναν; μα; 是Kähler当且仅当Γμ; ναvμν; μαvμ; 是弱Kähler当且仅当Gαμ; ναν; μα)vμ=0.

定理2[8] 设 (M, F) 为复Finsler流形, 则其为复Berwald当且仅当 (M, F) 为Kähler且为弱的复Berwald度量.

这里, 复Berwald指Γβ; μαβ; μα(z), 与纤维坐标无关; 弱的复Berwald度量指$ {{{\dot{\partial }}}_{{\bar{\alpha }}}}{{G}^{\beta }}=0 $[8-9].

2 测地线

由文献[7], 复Finsler流形上的测地线需满足:

$ {\nabla _{{T^H} + \overline {{T^H}} }}{T^H} = {\theta ^ * }\left( {{T^H},\overline {{T^H}} } \right), $ (3)

其中$ {{\theta }^{*}}={{G}^{\bar{\tau }\alpha }}{{G}_{\beta \bar{v}}}(\Gamma _{_{\bar{\mu };\bar{\tau }}}^{{\bar{v}}}-\Gamma _{_{\bar{\tau };\bar{\mu }}}^{{\bar{v}}})\text{d}{{z}^{\beta }}\wedge \text{d}{{z}^{{\bar{\mu }}}}\otimes {{\delta }_{\alpha }} $.文献[10]证明了$ {{\theta }^{*\alpha }}=2{{G}^{\bar{\tau }\alpha }}({{\overset{c}{\mathop{\delta }}\, }_{{\bar{\tau }}}}G)$, 其中$ {{\overset{c}{\mathop{\delta }}\, }_{\tau }}:={{\partial }_{\tau }}-({{l}_{\tau }}{{G}^{\alpha }}){{{\dot{\partial }}}_{\alpha }} $, 并且θ*α=0当且仅当 (M, F) 为弱Kähler.

从而,测地线σ=σ(s) 需满足:

$ {{\ddot \sigma }^\alpha }\left( s \right) + 2{G^\alpha }\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) - {\theta ^{ * \alpha }}\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = 0. $ (4)

注1 θ*α为 (1, 1) 齐次,即对于任意的ξ∈ℂ, θ*α(z, ξv)=ξξθ*α(z, v);

注2 可证明$ {{G}_{\alpha \bar{\beta }}}{{\theta }^{*\alpha }}\overline{{{\nu }^{\beta }}}=0 $.从而若令$ \theta ={{\theta }^{*\alpha }}{{{\dot{\partial }}}_{\alpha }} $, vVv的垂直提升[7], 则〈θ, vV〉=0, 即θ⊥vV.

定理3 复Finsler流形 (M, F) 上, 设σ=σ(s) 为测地线, 参数变换s=s($ {\tilde{s}} $), 则$\tilde{\sigma }\left( {\tilde{s}} \right)=\sigma \left( s\left( {\tilde{s}} \right) \right) $也为测地线的充要条件是${\tilde{s}} $=as+b(a, bR为常数), 即测地线参数相差一个仿射变换.

证明 充分性.在局部坐标系下, 有

$ \begin{array}{*{35}{l}} {{\sigma }^{\alpha }}\left( s \right)={{{\tilde{\sigma }}}^{\alpha }}\left( {\tilde{s}} \right),\ \ \ \ {{{\dot{\sigma }}}^{\alpha }}\left( s \right)={{{\dot{\tilde{\sigma }}}}^{\alpha }}\left( {\tilde{s}} \right)\frac{\text{d}\tilde{s}}{\text{d}s}, \\ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)={{{\ddot{\tilde{\sigma }}}}^{\alpha }}\left( {\tilde{s}} \right){{\left( \frac{\text{d}\tilde{s}}{\text{d}s} \right)}^{2}}={{{\dot{\tilde{\sigma }}}}^{\alpha }}\left( {\tilde{s}} \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}, \\ \end{array} $

因此,

$ \begin{array}{*{35}{l}} \left[ {{{\ddot{\tilde{\sigma }}}}^{\alpha }}\left( s \right)+2{{G}^{\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right)-{{\theta }^{*\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right) \right]{{\left( \frac{\text{d}\tilde{s}}{\text{d}s} \right)}^{2}}= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)-{{{\dot{\sigma }}}^{\alpha }}\left( {\tilde{s}} \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}+2{{G}^{\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right)\frac{\text{d}\tilde{s}}{\text{d}s} \right)- \\ \ \ \ \ \ \ \ \ {{\theta }^{*\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right)\left( \frac{\text{d}\tilde{s}}{\text{d}s} \right) \right)= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)+2{{G}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-{{\theta }^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)- \\ \ \ \ \ \ \ \ \ {{{\dot{\sigma }}}^{\alpha }}\left( s \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}}=-{{{\dot{\sigma }}}^{\alpha }}\left( s \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}}=0. \\ \end{array} $

注意到σ(s) 为正则曲线, 则必存在α, 使得${{{\dot{\sigma }}}^{\alpha }}\left( s \right)\ne 0 $, 又因为$ \frac{\text{d}\tilde{s}}{\text{d}s}\ne 0$, 所以由上式可得$ \frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\ne 0 $, 于是有$ \tilde{s}=as+b $(a, bRa≠0).

必要性显然.

3 射影等价及仿射等价

定义3 设F, $ {\tilde{F}} $均为M上的复Finsler度量, 若它们的测地线作为点集是相同的, 则称F, $ {\tilde{F}} $射影等价.即对于F的任意测地线σ=σ(s), 经适当的参数变换s=s($ {\tilde{s}}$) 后, 得到$ \tilde{\sigma }=\sigma \left( s\left( {\tilde{s}} \right) \right)=\tilde{\sigma }\left( {\tilde{s}} \right) $${\tilde{F}} $的测地线, 反之亦然.

定理4 复流形M上复Finsler度量F, ${\tilde{F}} $射影等价当且仅当$ {\tilde{M}} $上存在光滑实值函数P(z, v), 使得

$ {{\tilde G}^\alpha } = {G^\alpha } + {B^\alpha } + P{v^\alpha }, $

其中,$ {{{\tilde{G}}}^{_{\alpha }}} $, Gα分别为$ {\tilde{F}}$, F的测地系数, $ {{B}^{\alpha }}=\frac{1}{2}({{{\tilde{\theta }}}^{*\alpha }}-{{\theta }^{*\alpha }}) $.

证明 充分性.若F, $ {\tilde{F}} $射影等价,则对于 (M, $ {\tilde{F}} $) 上任意测地线σ=σ(s), 经参数变换s=s(${\tilde{s}} $) 得到的$ \tilde{\sigma }=\sigma \left( s\left( {\tilde{s}} \right) \right)=\tilde{\sigma }\left( {\tilde{s}} \right) $也是 (M, $ {\tilde{F}} $) 的测地线.在局部坐标系下, 有

$ \begin{array}{*{35}{l}} \left[ {{{\ddot{\tilde{\sigma }}}}^{\alpha }}\left( s \right)+2{{{\tilde{G}}}^{\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right)-{{{\tilde{\theta }}}^{*\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right) \right]{{\left( \frac{\text{d}\tilde{s}}{\text{d}s} \right)}^{2}}= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)-{{{\dot{\tilde{\sigma }}}}^{\alpha }}\left( {\tilde{s}} \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}+2{{{\tilde{G}}}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)- \\ \ \ \ \ \ \ \ \ {{{\tilde{\theta }}}^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)= \\ \ \ \ \ \ \ \ \ -2{{G}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)+{{\theta }^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)- \\ \ \ \ \ \ \ \ \ {{{\dot{\sigma }}}^{\alpha }}\left( s \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}}+2{{{\tilde{G}}}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)- \\ \ \ \ \ \ \ \ \ {{{\tilde{\theta }}}^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)=0. \\ \end{array} $

因此,

$ \begin{array}{l} {{\tilde G}^\alpha }\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = {G^\alpha }\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) + \\ \;\;\;\;\;\;\;\;\frac{1}{2}\left[ {{{\tilde \theta }^{ * \alpha }}\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) - {\theta ^{ * \alpha }}\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right)} \right] + \\ \;\;\;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {\frac{1}{2}}&{\frac{{{{\rm{d}}^2}\tilde s}}{{{\rm{d}}{s^2}}}}&{\frac{1}{{\frac{{{\rm{d}}\tilde s}}{{{\rm{d}}s}}}}} \end{array}} \right){{\dot \sigma }^\alpha }\left( s \right). \end{array} $

σ(s), $ \dot{\sigma }\left( s \right) $的任意性, 有

$ {{\tilde G}^\alpha } = {G^\alpha } + {B^\alpha } + P{v^\alpha }, $

其中, $ P\left( z, v \right)=\frac{1}{2}\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}} $为一阶实值函数.

必要性.若$ {{{\tilde{G}}}^{\alpha }}={{G}^{\alpha }}+{{B}^{\alpha }}+P{{v}^{\alpha }} $, P(z, v) 为光滑实值函数, 那么对于 (M, F) 上任意测地线σ=σ(s), 经参数变换为$ {\tilde{\sigma }} $=$ {\tilde{\sigma }} $(${\tilde{s}} $), 即可得

$ \begin{array}{*{35}{l}} \left[ {{{\ddot{\tilde{\sigma }}}}^{\alpha }}\left( {\tilde{s}} \right)+2{{{\tilde{G}}}^{\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right)-{{{\tilde{\theta }}}^{*\alpha }}\left( \tilde{\sigma }\left( {\tilde{s}} \right),\dot{\tilde{\sigma }}\left( {\tilde{s}} \right) \right) \right]{{\left( \frac{\text{d}\tilde{s}}{\text{d}s} \right)}^{2}}= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)+2{{{\tilde{G}}}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)- \\ \ \ \ \ \ \ \ \ {{{\tilde{\theta }}}^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-\dot{\sigma }\left( s \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}}= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)+2{{G}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)+ \\ \ \ \ \ \ \ \ \ 2{{B}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-{{{\tilde{\theta }}}^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)+ \\ \ \ \ \ \ \ \ \ P\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right){{{\dot{\sigma }}}^{\alpha }}\left( s \right)-{{{\dot{\sigma }}}^{\alpha }}\left( s \right)\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}}= \\ \ \ \ \ \ \ \ \ {{{\ddot{\sigma }}}^{\alpha }}\left( s \right)+2{{G}^{\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-{{\theta }^{*\alpha }}\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)+ \\ \ \ \ \ \ \ \ \ \left[ P\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}} \right]{{{\dot{\sigma }}}^{\alpha }}\left( s \right)= \\ \ \ \ \ \ \ \ \ \left[ P\left( \sigma \left( s \right),\dot{\sigma }\left( s \right) \right)-\frac{{{\text{d}}^{2}}\tilde{s}}{\text{d}{{s}^{2}}}\frac{1}{\frac{\text{d}\tilde{s}}{\text{d}s}} \right]{{{\dot{\sigma }}}^{\alpha }}\left( s \right). \\ \end{array} $

$ {\tilde{\sigma }} $(${\tilde{s}} $) 为 (M, $ {\tilde{F}} $) 的测地线, 则

$ \dot \sigma \left( s \right) = 0. $

σ(s) 的正则性, 得

$ P\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = \frac{{{{\rm{d}}^2}\tilde s}}{{{\rm{d}}{s^2}}}\frac{1}{{\frac{{{\rm{d}}\tilde s}}{{{\rm{d}}s}}}}, $

$t=\frac{\text{d}\tilde{s}}{\text{d}s} $, 则

$ P\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right){\rm{d}}s = \frac{{{\rm{d}}t}}{t}, $

等式两边积分,有

$ t = \frac{{{\rm{d}}\tilde s}}{{{\rm{d}}s}} = a{{\rm{e}}^{\int {P\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right){\rm{d}}s} }}, $

最终得到

$ \tilde s = a\int {{{\rm{e}}^{\int {P\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right){\rm{d}}s} }}{\rm{d}}s} + b, $

其中a, bR为常数.从而F, $ {\tilde{F}} $射影等价.

定理5 复流形M上复Finsler度量F, $ {\tilde{F}} $射影等价当且仅当存在$ {\tilde{M}} $上的光滑函数R(z, v), Q(z, v), 使得Bα=Rvα, $ {{{\tilde{G}}}^{\alpha }}={{G}^{\alpha }}+Q{{v}^{\alpha }} $, 且P(z, v)=Q(z, v)-R(z, v) 为实值函数.

证明 由定义可知,$ {{{\tilde{G}}}^{\alpha }} $, Gα为 (2, 0) 齐次, Bα为 (1, 1) 齐次.

充分性.由$ {{B}^{\alpha }}={{v}^{{\bar{\tau }}}}B_{\tau }^{^{\alpha }}=\left(-{{v}^{^{{\bar{\tau }}}}}{{P}_{^{{\bar{\tau }}}}} \right){{v}^{\alpha }} $, 得$ R=-{{v}^{^{{\bar{\tau }}}}}{{P}_{^{{\bar{\tau }}}}} $, 结论成立.

必要性显然.

注3 由定理5知,Q(z, v) 为 (1, 0) 齐次, R(z, v) 为 (0, 1) 齐次, 所以Qαvα=Q, $ {{Q}_{^{{\bar{\alpha }}}}}{{v}^{{\bar{\alpha }}}}=0 $, Rαvα=0, $ {{R}_{^{{\bar{\alpha }}}}}{{v}^{{\bar{\alpha }}}}=R $, 从而$ {{P}_{\alpha }}{{v}^{\alpha }}+{{P}_{{\bar{\alpha }}}}{{v}^{{\bar{\alpha }}}}=P={{P}_{a}}{{u}^{a}} $, 即P为一阶齐次实值函数, 且可以得到$ R=-{{v}^{{\bar{\alpha }}}}{{P}_{{\bar{\alpha }}}} $, Q=Pαvα.

定理6[5] 复流形M上复Finsler度量F, $ {\tilde{F}} $射影等价当且仅当存在$ {\tilde{M}} $上光滑一阶齐次实值函数P(z, υ), 使得

$ \begin{array}{*{20}{c}} {{B^\alpha } = - {v^{\bar \tau }}{P_{\bar \tau }}{v^\alpha },}\\ {{{\tilde G}^\alpha } = {G^\alpha } + \left( {{v^\tau }{P_\tau }} \right){v^\alpha }.} \end{array} $ (5)

注4 若一阶齐次函数P1(z, υ) 满足式 (5), S为 (0, 1) 阶齐次, 则P2=P1+S也为一阶齐次, 且P2亦满足式 (5);反之, 若一阶齐次函数P1(z, υ), P2(z, υ) 满足式 (5), 则S=P2-P1为 (0, 1) 阶齐次.故满足式 (5) 的一阶齐次解相差一个 (0, 1) 阶齐次函数.

定理7 复流形M上复Finsler度量F, $ {\tilde{F}} $相应的测地系数分别为Gα, $ {{{\tilde{G}}}^{\alpha }} $, 则

$ {{\tilde G}^\alpha } = {G^\alpha } + R{v^\alpha } + {Q^\alpha }, $ (6)

其中, $ R=\frac{{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}}{2\tilde{F}} $, $ {{Q}^{\alpha }}=\tilde{F}{{{\tilde{G}}}^{\bar{v}\alpha }}[{{\nu }^{\tau }}{{{\tilde{F}}}_{|\tau \bar{v}}}+2{{({{G}^{\tau }})}_{{\bar{v}}}}{{{\tilde{F}}}_{\tau }}] $.

证明 因为$ {{{\tilde{G}}}_{{\bar{v}}}}=2\tilde{F}{{{\tilde{F}}}_{{\bar{v}}}} $, $ {{{\tilde{G}}}_{\mu \bar{v}}}=2({{{\tilde{F}}}_{\mu }}{{{\tilde{F}}}_{{\bar{v}}}}+\tilde{F}{{{\tilde{F}}}_{\mu \bar{v}}}) $,且由齐次性, 有

$ \begin{array}{l} {{\tilde G}^{\bar v\alpha }}{{\tilde G}_{\bar v}} = {{\tilde G}^{\bar v\alpha }}{{\tilde G}_{\mu \bar \nu }}{v^\mu } = {v^\alpha },\;\;\;\;\;{{\tilde G}^{\bar v\alpha }}{{\tilde F}_{\bar \nu }} = \frac{{{v^\alpha }}}{{2\tilde F}},\\ {{\tilde G}^{\bar v\alpha }}{{\tilde G}_{\mu \bar \nu }} = {{\tilde G}^{\bar v\alpha }}\frac{{{{\tilde G}_{\mu \bar \nu }} - 2{{\tilde F}_\mu }{{\tilde F}_{\bar \nu }}}}{{2\tilde F}} = \frac{{\tilde F{\delta _{\alpha \mu }} - {v^\alpha }{{\tilde F}_\mu }}}{{2{{\tilde F}^2}}}, \end{array} $

所以,

$ \begin{array}{l} {{\tilde G}^\alpha } = \frac{1}{2}\tilde \Gamma _{;\tau }^\alpha {v^\tau } = \frac{1}{2}{{\tilde G}^{\bar v\alpha }}{{\tilde G}_{\bar \nu ;\tau }}{v^\tau } = \\ \;\;\;\;\;\;\;\;\frac{1}{2}{{\tilde G}^{\bar v\alpha }} \cdot 2\left( {{{\tilde F}_{;\tau }}{{\tilde F}_{\bar \nu }} + \tilde F{{\tilde F}_{\bar \nu ;\tau }}} \right){v^\tau } = \\ \;\;\;\;\;\;\;\;\frac{{{v^\tau }{{\tilde F}_{;\tau }}}}{{2\tilde F}}{v^\alpha } + \tilde F{{\tilde G}^{\bar v\alpha }}{{\tilde F}_{\bar \nu ;\tau }}{v^\tau }. \end{array} $ (7)

通过计算$ {\tilde{F}} $的水平导数, 可求得式 (7) 的后一项,

$ {v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} = {v^\tau }{{\tilde F}_{\bar \nu ;\tau }} - 2{\left( {{G^\tau }} \right)_{\bar \nu }}{{\tilde F}_\tau } - {v^\tau }\tilde \Gamma _{;\tau }^\mu {{\tilde F}_{\mu \bar \nu }}, $

${{{\tilde{G}}}^{\bar{v}\alpha }} $缩并, 得到

$ \begin{array}{*{20}{c}} {{{\tilde G}^{\bar v\alpha }}{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} = {{\tilde G}^{\bar v\alpha }}{v^\tau }{{\tilde F}_{\bar \nu ;\tau }} - 2{{\tilde G}^{\bar v\alpha }}{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau } - }\\ {{G^\mu }\frac{{{\delta _{\mu \alpha }} - \frac{{{v^\alpha }{{\tilde F}_\mu }}}{{\tilde F}}}}{{\tilde F}},} \end{array} $ (8)

式 (8) 乘以$ {\tilde{F}} $, 得到

$ \begin{array}{*{20}{c}} {\tilde F{{\tilde G}^{\bar v\alpha }}{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} = \tilde F{{\tilde G}^{\bar v\alpha }}{v^\tau }{{\tilde F}_{\bar \nu ;\tau }} - 2\tilde F{{\tilde G}^{\bar v\alpha }}{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau } - }\\ {{G^\alpha } + \frac{{{G^\tau }{{\tilde F}_\tau }{v^\alpha }}}{{\tilde F}},} \end{array} $ (9)

将式 (9) 代入式 (7), 可得

$ \begin{array}{*{20}{c}} {\tilde F{{\tilde G}^{\bar v\alpha }}{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} = {{\tilde G}^\alpha } - \frac{{{{\tilde F}_{;\tau }}{v^\tau }}}{{2\tilde F}}{v^\alpha } - 2\tilde F{{\tilde G}^{\bar v\alpha }}{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau } - }\\ {{G^\alpha } + \frac{{{G^\tau }{{\tilde F}_\tau }{v^\alpha }}}{{\tilde F}}.} \end{array} $ (10)

因此,

$ \begin{array}{l} {{\tilde G}^\alpha } = {G^\alpha } + \left( {\frac{{{{\tilde F}_{;\tau }}{v^\tau }}}{{2\tilde F}} - \frac{{{G^\tau }{{\tilde F}_\tau }}}{{\tilde F}}} \right){v^\alpha } + \\ \;\;\;\;\;\;\;\;\;\tilde F{{\tilde G}^{\bar v\alpha }}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right], \end{array} $ (11)

而且,

$ \frac{{{{\tilde F}_{;\tau }}{v^\tau }}}{{2\tilde F}} - \frac{{{G^\tau }{{\tilde F}_\tau }}}{{\tilde F}} = \frac{{{{\tilde F}_{;\tau }}{v^\tau } - \Gamma _{;\tau }^\nu {v^\tau }{F_\nu },}}{{2\tilde F}} = \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}, $

代入式 (11), 定理7得证.

反复利用式 (6) 可得:

定理8 复流形M上复Finsler度量F, $ {\tilde{F}} $,以下条件等价:

1) 存在$ {\tilde{M}} $上一阶齐次函数P(z, v), 使得

$ {{\tilde G}^\alpha } = {G^\alpha } + \left( {{v^\tau }{P_\tau }} \right){v^\alpha }; $

2) $ {{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}+2{{({{G}^{\tau }})}_{{\bar{v}}}}{{{\tilde{F}}}_{\tau }}=\frac{1}{{\tilde{F}}}{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}{{{\tilde{F}}}_{{\bar{v}}}} $;

3) $ {{{\tilde{G}}}^{\alpha }}={{G}^{\alpha }}+\frac{{{v}^{\tau }}{{{\tilde{F}}}_{\tau }}}{{\tilde{F}}}{{v}^{\alpha }} $.

证明 1)⇒2)

$ \left( {{v^\tau }{P_\tau }} \right){v^\alpha } = \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } + \tilde F{{\tilde G}^{\bar v\alpha }}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right], $ (12)

$ {{{\tilde{G}}}_{\bar{\mu }\alpha }} $缩并, 于是有

$ \left( {{v^\tau }{P_\tau }} \right){{\tilde G}_{\bar \mu }} = \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{{\tilde G}_{\bar \mu }} + \tilde F\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right], $

再与vμ缩并, 于是有

$ \left( {{v^\tau }{P_\tau }} \right){{\tilde F}^2} = \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{2}\tilde F + \tilde F \cdot \frac{1}{2}{v^\tau }{{\tilde F}_{\left| \tau \right.}} = \tilde F{v^\tau }{{\tilde F}_{\left| \tau \right.}}, $

$ {{v}^{\tau }}{{P}_{\tau }}=\frac{{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}}{{\tilde{F}}} $, 代入式 (12), 得

$ \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } = \tilde F{{\tilde G}^{\bar v\alpha }}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right]. $

因此,有

$ {v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{\left( {{G^\tau }} \right)_{\bar \nu }}{{\tilde F}_\tau } = \frac{1}{{\tilde F}}{v^\tau }{{\tilde F}_{\left| \tau \right.}}{{\tilde F}_{\bar v}}. $

2) ⇒3)

$ \begin{array}{l} \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } + \tilde F{{\tilde G}^{\bar v\alpha }}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right] = \\ \;\;\;\;\;\;\;\frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } + \tilde F{{\tilde G}^{\bar v\alpha }}\frac{1}{{\tilde F}}{v^\tau }{{\tilde F}_{\left| \tau \right.}}{{\tilde F}_{\bar v}} = \\ \;\;\;\;\;\;\;\frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } + {v^\tau }{{\tilde F}_{\left| \tau \right.}}\frac{{{v^\alpha }}}{{2\tilde F}} = \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{\tilde F}}{v^\alpha }. \end{array} $

3) ⇒1)

$ P=\frac{{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}+{{\theta }^{*\tau }}{{{\tilde{F}}}_{\tau }}}{{\tilde{F}}} $, 则由于$ {{v}^{\tau }}\tilde{F} $为 (1, 0) 齐次, 有$ \frac{{{\theta }^{*\tau }}{{{\tilde{F}}}_{\tau }}}{{\tilde{F}}} $为 (0, 1) 齐次, $ $, 所以$ {{{\tilde{G}}}^{\alpha }} $=Gα+(υτPτ)vα, 且vτPτ+vτPτ=P为一阶齐次.证毕.

定理9 复流形M上复Finsler度量F, $ {\tilde{F}} $射影等价当且仅当$ {{B}^{\alpha }}=-\frac{{{\theta }^{*\tau }}{{{\tilde{F}}}_{\tau }}}{{\tilde{F}}}{{\upsilon }^{\alpha }} $, 且$ {{{\tilde{G}}}^{\alpha }}={{G}^{\alpha }}+\frac{{{v}^{\tau }}{{F}_{\tau }}}{F}{{v}^{\alpha }} $.

定义4 复流形M上复Finsler度量F, $ {\tilde{F}} $, 若它们有相同的测地线, 则称F, $ {\tilde{F}} $仿射等价.即对F的任意测地线σ=σ(s) 也必为$ {\tilde{F}} $的测地线, 反之亦然.

定理10[11] 复流形M上弱Kähler-Finsler度量F, $ {\tilde{F}} $,若其射影等价,则必有仿射等价.此时$ {{{\tilde{G}}}^{\alpha }} $=Gα.

由文献[5]知, 复Finsler度量F, $ {\tilde{F}} $射影等价, 若$ {\tilde{F}} $为弱Kähler, 则F也必为弱Kähler.故定理10只须假定F, $ {\tilde{F}} $其中之一为弱Kähler, 且仅需研究仿射等价.

定理11 复流形M上弱Kähler-Finsler度量F, $ {\tilde{F}} $, 则F, $ {\tilde{F}} $仿射等价当且仅当

$ {v^\tau }{{\tilde F}_{\left| \tau \right.}} = 0\;\;且\;{\left( {{G^\tau }} \right)_{\bar \nu }}{{\tilde F}_\tau } = 0. $

证明 充分性.若F, $ {\tilde{F}} $仿射等价, 则由定理10可知$ {{{\tilde{G}}}^{\alpha }} $=Gα, 即

$ \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{{2\tilde F}}{v^\alpha } + \tilde F{{\tilde G}^{\bar v\alpha }}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right] = 0, $ (13)

$ {{{\tilde{G}}}_{\alpha }} $缩并, 有

$ \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{2}\tilde F + \tilde F{v^{\bar v}}\left[ {{v^\tau }{{\tilde F}_{\left| {\tau \bar v} \right.}} + 2{{\left( {{G^\tau }} \right)}_{\bar \nu }}{{\tilde F}_\tau }} \right] = 0. $

Gτ为 (2, 0) 齐次, 可知

$ \frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{2}\tilde F + \tilde F\frac{{{v^\tau }{{\tilde F}_{\left| \tau \right.}}}}{2} = 0, $

从而$ {{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}=0 $, 代入式 (13), 可得$ {{({{G}^{\tau }})}_{{\bar{v}}}}{{{\tilde{F}}}_{\tau }}=0 $.

必要性显然.

结合定理10和定理11,可以得到以下结论:

定理12 对于复流形M上的弱Kähler-Finsler度量F, $ {\tilde{F}} $, 以下条件等价:

(1) F, $ {\tilde{F}} $射影等价;

(2) F, $ {\tilde{F}} $仿射等价;

(3) $ {{{\tilde{G}}}^{\alpha }} $=Gα;

(4) $ {{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}=0 $$ {{({{G}^{\tau }})}_{{\bar{v}}}}{{{\tilde{F}}}_{\tau }}=0 $.

定理13 对于复流形M上的弱Kähler-Finsler度量F, $ {\tilde{F}} $, 若F为弱的复Berwald度量, 则F, $ {\tilde{F}} $仿射等价当且仅当$ {{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}=0 $.

4 平行移动

如不特别说明, 下面复Finsler流形均指弱Kähler, 其测地线σ=σ(s) 满足二阶微分方程

$ {{\ddot \sigma }^\alpha }\left( s \right) + 2{G^\alpha }\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = 0. $

定义5 复Finsler流形 (M, F) 上, σ=σ(s) 为光滑正则曲线, U=Uα(s)α|σ(s)为沿着σ定义的向量场, 则U(s) 沿着σ的线性共变导数:

$ {D_{\dot \sigma }}U\left( s \right) = \left[ {{{\dot U}^\alpha }\left( s \right) + {U^\tau }\left( s \right)\Gamma _{;\tau }^\alpha \left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right)} \right]{\partial _\alpha }\left| {_{\sigma \left( s \right)}} \right.. $ (14)

可以证明: $ {{D}_{{\dot{\sigma }}}}\left( U+V \right)\left( s \right)={{D}_{{\dot{\sigma }}}}U\left( s \right)+{{D}_{{\dot{\sigma }}}}V\left( s \right) $,

$ {D_{\dot \sigma }}\left( {f\left( U \right)} \right)\left( s \right) = f'\left( s \right)U\left( s \right) + f\left( s \right){D_{\dot \sigma }}U\left( s \right). $

定义6 设U=U(S) 是沿着σ的向量场, 若$ {{D}_{{\dot{\sigma }}}}U\left( s \right)=0 $, 则称U=U(S) 沿着σ线性平行.

由定义可知,σ为测地线当且仅当$ \dot{\sigma }(s) $沿着σ线性平行.

定理14 若 (M, F) 为Kähler-Finsler流形, σ=σ(s) 为测地线, U(s), V(s) 沿着σ线性平行, 则$ \langle U, V{{\rangle }_{\dot{\sigma }(s)}}:={{g}_{\dot{\sigma }(s)}}\left( U\left( s \right), V\left( s \right) \right):={{G}_{\alpha \bar{\beta }}}{{U}^{\alpha }}\ \overline{{{V}^{\beta }}}$为常数.

证明 令$ v=\dot{\sigma }\left( s \right) $, 由σ为测地线, 有$ {{{\ddot{\sigma }}}^{\alpha }}+\Gamma _{_{;\tau }}^{^{\alpha }}{{v}^{\tau }}=0 $.因此,

$ \begin{array}{l} \frac{{{\rm{d}}\left( {{G_{\alpha \bar \beta }}{U^\alpha }\overline {{V^\beta }} } \right)}}{{{\rm{d}}s}} = \left( {{G_{\alpha \bar \beta ;\tau }}{v^\tau } + {G_{\alpha \bar \beta ;\bar \tau }}{v^{\bar \tau }} + {G_{\alpha \bar \beta \tau }}{{\ddot \sigma }^\tau } + } \right.\\ \;\;\;\;\;\;\;\left. {{G_{\alpha \bar \beta \bar \tau }}\overline {{{\ddot \sigma }^\tau }} } \right){U^\alpha }\overline {{V^\beta }} + {G_{\alpha \bar \beta }}{{\dot U}^\alpha }\overline {{V^\beta }} + {G_{\alpha \bar \beta }}{U^\alpha }\overline {{{\dot V}^\beta }} = \\ \;\;\;\;\;\;\;\left( {{G_{\alpha \bar \beta ;\tau }}{v^\tau } + {G_{\alpha \bar \beta ;\bar \tau }}\overline {{v^\tau }} - {G_{\alpha \bar \beta \tau }}\Gamma _{;\mu }^\tau {v^\mu } - } \right.\\ \;\;\;\;\;\;\;\left. {{G_{\alpha \bar \beta \bar \tau }}\Gamma _{;\bar \mu }^{\bar \tau }\overline {{v^\mu }} } \right){U^\alpha }\overline {{V^\beta }} - {G_{\alpha \bar \beta }}\Gamma _{;\mu }^\alpha {U^\tau }\overline {{V^\beta }} - {G_{\alpha \bar \beta }}\Gamma _{;\tau }^{\bar \beta }{U^\alpha }\overline {{V^\tau }} = \\ \;\;\;\;\;\;\;\left[ {\left( {{G_{\alpha \bar \beta ;\mu }} - {G_{\alpha \bar \beta \tau }}\Gamma _{;\mu }^\tau } \right){v^\mu } + \overline {\left( {{G_{\beta \bar \alpha ;\mu }} - {G_{\beta \bar \alpha \tau }}\Gamma _{;\mu }^\tau } \right){v^\mu }} } \right. - \\ \;\;\;\;\;\;\;\left. {{G_{\tau \bar \beta }}\Gamma _{;\alpha }^\tau - \overline {{G_{\tau \bar \alpha }}\Gamma _{;\beta }^\tau } } \right]{U^\alpha }\overline {{V^\beta }} . \end{array} $

$ \Gamma _{_{\alpha ;\mu }}^{^{\tau }}={{G}^{\bar{\beta }\tau }}({{G}_{\alpha \bar{\beta };\mu }}-{{G}_{\alpha \bar{\beta }\nu }}\Gamma _{_{;\mu }}^{^{\nu }}) $, 则有

$ {G_{\alpha \bar \beta ;\mu }} - {G_{\alpha \bar \beta \nu }}\Gamma _{;\mu }^\tau = {G_{\tau \bar \beta }}\Gamma _{\alpha ;\mu }^\tau . $

注意到F为Kähler, 则

$ \left( {{G_{\alpha \bar \beta ;\mu }} - {G_{\alpha \bar \beta \nu }}\Gamma _{;\mu }^\tau } \right){v^\mu } = {G_{\tau \bar \beta }}\Gamma _{\alpha ;\mu }^\tau {v^\mu } = {G_{\tau \bar \beta }}\Gamma _{\mu ;\alpha }^\tau {v^\mu } = {G_{\tau \bar \beta }}\Gamma _{;\alpha }^\tau . $

因此$ \frac{\text{d}\langle U, V{{\rangle }_{\dot{\sigma }(s)}}}{\text{d}s}=0 $, 即$ \langle U, V{{\rangle }_{\dot{\sigma }(s)}} $为常数.

定义7 弱Kähler-Finsler流形 (M, F) 上, 设σ=σ(s) 为光滑正则曲线, U=Uα(s)α|σ(s)是沿着σ定义的向量场, 那么U(s) 沿着σ的共变导数:

$ {\nabla _{\dot \sigma }}U\left( s \right) = \left[ {{{\dot U}^\alpha }\left( s \right) + \Gamma _{;\tau }^\alpha \left( {\sigma \left( s \right),\dot U\left( s \right)} \right) \cdot {{\dot \sigma }^\tau }\left( s \right)} \right]{\partial _\alpha }\left| {_{\dot \sigma \left( s \right)}} \right., $ (15)

$ {{\nabla }_{{\dot{\sigma }}}} $U一般不具有线性.

定义8 设U=U(s) 为沿着σ的向量场, 若$ {{\nabla }_{{\dot{\sigma }}}}U(s)=0 $, 则称U=U(s) 沿着σ平行.

同样由定义可知,σ为测地线当且仅当$\dot{\sigma }\left( s \right) $沿着σ平行.

定理15 复Finsler流形 (M, F) 上, 设σ=σ(s) 为测地线, U(s) 沿着σ平行, 则F(σ(s), U(s)) 为常数.特别地,F(σ(s), $ \dot{\sigma }\left( s \right) $) 是一个常数.

证明 由于

$ {F^2}\left( {\sigma \left( s \right),U\left( s \right)} \right) = {G_{\alpha \bar \beta }}\left( {\sigma \left( s \right),U\left( s \right)} \right){U^\alpha }\left( s \right)\overline {{U^\beta }\left( s \right)} , $

因此,

$ \begin{array}{*{20}{c}} {\frac{{\rm{d}}}{{{\rm{d}}s}}{F^2}\left( {\sigma \left( s \right),U\left( s \right)} \right) = \left( {{G_{\alpha \bar \beta ;\tau }}{{\dot \sigma }^\tau } + {G_{\alpha \bar \beta ;\bar \tau }}\overline {{{\dot \sigma }^\tau }} + {G_{\alpha \bar \beta \tau }}{{\ddot \sigma }^\tau } + } \right.}\\ {\left. {{G_{\alpha \bar \beta \bar \tau }}\overline {{{\ddot \sigma }^\tau }} } \right){U^\alpha }\overline {{U^\beta }} + {G_{\alpha \bar \beta }}{{\dot U}^\alpha }\overline {{U^\beta }} + {G_{\alpha \bar \beta }}{U^\alpha }\overline {{{\dot U}^\beta }} .} \end{array} $

注意到

$ \begin{array}{*{20}{c}} {{G_{\alpha \bar \beta \tau }}{U^\alpha } = 0,\;\;\;\;\;{G_{\alpha \bar \beta \bar \tau }}\overline {{U^\beta }} = 0,}\\ {{G_{\alpha \bar \beta ;\tau }}{U^\alpha }\overline {{U^\beta }} = {G_{;\tau }},\;\;\;\;\;{G_{\alpha \bar \beta ;\bar \tau }}{U^\alpha }\overline {{U^\beta }} = {G_{;\bar \tau }}.} \end{array} $

经化简, 可得

$ \frac{{{\rm{d}}{F^2}}}{{{\rm{d}}s}} = {G_{;\tau }}{{\dot \sigma }^\tau } + \overline {{G_{;\tau }}{{\dot \sigma }^\tau }} - {G_\alpha }\Gamma _{;\tau }^\alpha {{\dot \sigma }^\tau } - {G_{\bar \beta }}\Gamma _{;\bar \tau }^{\bar \beta }\overline {{{\dot \sigma }^\tau }} . $

又因为

$ {G_\alpha }\Gamma _{;\tau }^\alpha = {G_\alpha }{G^{\bar \nu \alpha }}{G_{\bar \nu ;\tau }} = {G_{;\tau }}, $

$ \frac{{{\rm{d}}{F^2}}}{{{\rm{d}}s}} = 0, $

F(σ(s), U(s)) 为常数.

定理16 若F, $ {\tilde{F}} $为复流形M上的弱Kähler-Finsler度量, 且F为弱的复Berwald度量, 则F$ {\tilde{F}} $仿射等价当且仅当对于 (M, F) 上的任意测地线σ, $ {\tilde{F}} $(σ(s), $\dot{\sigma }\left( s \right) $) 为常数.

证明 充分性.对于 (M, F) 上任意测地线σ, 有$ {{{\ddot{\sigma }}}^{\alpha }}+\Gamma _{_{;\tau }}^{^{\alpha }}{{{\dot{\sigma }}}^{\tau }}=0, v=\dot{\sigma } $.利用定理13, 有

$ \begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}s}}\tilde F\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = {{\tilde F}_{;\tau }}{{\dot \sigma }^\tau } + {{\tilde F}_{;\bar \tau }}{{\dot \sigma }^{\bar \tau }} + {{\tilde F}_\tau }{{\ddot \sigma }^\tau } + {{\tilde F}_\tau }\overline {{{\ddot \sigma }^\tau }} = \\ \;\;\;\;\;\;\;\;{{\tilde F}_{;\tau }}{v^\tau } - {{\tilde F}_\tau }\Gamma _{;\nu }^\tau {v^\nu } + \overline {{{\tilde F}_{;\tau }}{v^\tau } - {{\tilde F}_\tau }\Gamma _{;\nu }^\tau {v^\nu }} = \\ \;\;\;\;\;\;\;\;{v^\tau }\tilde F\left| {_\tau } \right. + \overline {{v^\tau }\tilde F\left| {_\tau } \right.} = 0, \end{array} $

$ {\tilde{F}} $(σ(s), $ {\dot{\sigma }(s)}$) 为常数.

必要性.若$ {\tilde{F}} $(σ(s), $ {\dot{\sigma }(s)}$) 为常数, 有$ {{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}+\overline{{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}}=0 $.令$ Q\left( z, v \right)=\frac{{{v}^{\tau }}{{{\tilde{F}}}_{|\tau }}}{{\tilde{F}}} $, 则Q(z, v) 为 (1, 0) 齐次, 且Q+Q=0, Q(z, v)=iS(z, v), 其中i=$ \sqrt{-1} $, S(z, v)∈R.由此可知, S(z, v) 也为 (1, 0) 齐次,且S(z, iv)=iS(z, v)∈R, 于是有S(z, v)=0, Q(z, v)=0, ${{v}^{\tau }}{{{\tilde{F}}}_{|\tau }} $, 再由σ(s), $ \dot{\sigma }(s) $的任意性及定理13, 得F, $ {\tilde{F}} $仿射等价.

例1 设二元函数f:R2R+满足:对任意的λ>0, f(λs, λt)=λf(s, t), 且对于任意的 (s, t)≠(0, 0),f(s, t)>0.

又 (Mi, αi), i=1, 2为Hermitian度量, M=M1 ×M2, M1n维, M2m维复流形.可以构造新的度量

$ \tilde F\left( {z,v} \right) = \sqrt {f\left( {\alpha _1^2\left( {{z_1},{v_1}} \right),\alpha _2^2\left( {{z_2},{v_2}} \right)} \right)} , $

其中z=(z1, z2)∈M, 并且v=v1v2Tz1, 0M$ \cong $T1, 0z1M1T1, 0z2M2.

文献[12]证明了 (M, $ {\tilde{F}} $) 为强拟凸复Finsler流形的充要条件为f(s, t) 满足

$ \begin{array}{l} \;\;\;\;\;\;\;\;\;{f_s} > 0,{f_t} > 0,{f_s} + s{f_{ss}} > 0,{f_t} + t{f_u} > 0\\ 及\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f_s}{f_t} - f{f_{st}} > 0. \end{array} $

fsft-ffst>0.

$ F\left( z, v \right)=\sqrt{\alpha _{_{1}}^{^{2}}+\alpha _{_{2}}^{^{2}}} $, 则F也为Hermitian度量.经简单计算, 可得其测地系数是α1, α2测地系数的直积, 即

$ {G^a}\left( {z,v} \right) = {G^a}\left( {{z_1},{v_1}} \right),\;{G^a}\left( {z,v} \right) = {G^a}\left( {{z_2},{v_2}} \right), $

其中指标$a=\overline{1, n} $, $ \alpha =\overline{n+1, n+m} $.从而,σ=(σ1(s), σ2(s)) 为M测地线当且仅当σ1, σ2分别为M1, M2测地线, 故由定理15, αi(σ1(s), σ2(s)) 为常数, 从而

$ \tilde F\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right) = \sqrt {f\left( {\alpha _1^2\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right),\alpha _2^2\left( {\sigma \left( s \right),\dot \sigma \left( s \right)} \right)} \right)} $

为常数, 由定理16, $ {\tilde{F}} $F仿射等价, 且$ {{G}^{\alpha }}={{G}^{\alpha }}=\frac{1}{2}\Gamma _{_{\beta ;\tau }}^{^{\alpha }}\left( z \right){{v}^{\beta }}{{v}^{\tau }} $为Berwald度量.

参考文献
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