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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 0

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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.
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文章历史

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

1 预备知识

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).

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

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

2 测地线

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

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

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

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

3 射影等价及仿射等价

4 平行移动

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

F(σ(s), U(s)) 为常数.

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

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

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

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

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

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

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

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