AI 中文总结
研究有限维光滑流形上黄金芬斯勒结构，从局部和全局角度研究。局部计算相关张量等，研究射影平坦性；全局定义黄金芬斯勒变化，研究其几何性质及结构变换，最终证明\(\widetilde{F}\)与\(F\)不能射影相关。
AI 中文摘要
我们在有限维光滑流形\(M\)上引入黄金芬斯勒结构的概念，并从局部（基于坐标）和全局（无坐标）的角度对其进行研究。局部上，我们明确计算基本度量张量，建立正定条件，并推导测地线喷雾系数。此外，我们研究了黄金\((\alpha, \beta)\)-度量的射影平坦性，并证明了几乎有理的黄金\((\alpha, \beta)\)-度量不存在。全局上，我们定义了基础芬斯勒度量\(F\)的黄金芬斯勒变化\(\widetilde{F}\)，并利用特殊的共点\(\pi\)-向量场研究其几何性质。我们明确确定了包括巴特尔和贝尔瓦尔德联络在内的基本非线性结构在此变化下如何变换。最后，我们证明了\(\widetilde{F}\)和\(F\)不能射影相关。
英文摘要
We introduce the concept of a golden Finsler structure on a finite-dimensional smooth manifold $M$ and investigate it from both local (coordinate-based) and global (coordinate-free) perspectives. Locally, we explicitly compute the fundamental metric tensor, establish the positive definiteness condition, and derive the geodesic spray coefficients. Furthermore, we investigate the projective flatness of the golden $(α, β)$-metric and prove the non-existence of almost rational golden $(α, β)$-metrics. Globally, we define the golden Finsler change $\widetilde{F}$ of a base Finsler metric $F$ and examine its geometric properties utilizing a special concurrent $π$-vector field. We explicitly determine how fundamental non-linear structures, including the Barthel and Berwald connections, transform under this change. Finally, we prove that $\widetilde{F}$ and $F$ cannot be projectively related.
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