Finsler Geometry on Complex Vector Bundles
Finsler Geometry on Complex Vector Bundles
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发表时间：
2004
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通讯作者：
T. Aikou
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作者：
T. Aikou
流形或向量丛的Finsler度量被定义为每个纤维空间上每个基点的范数的光滑分配，因此Finsler度量类包含黎曼度量作为特殊子类。由于这个原因，芬斯勒几何通常被视为黎曼几何的推广。事实上，有许多贡献芬斯勒几何包含黎曼几何作为一个特殊情况（见例如，[Bao等人，2000]，[松本1986]，以及其中的参考文献）。另一方面，我们可以把芬斯勒几何当作黎曼几何的一个特例，因为芬斯勒几何可以发展为复流形的微分几何（例如，[Aikou 2002]）。事实上，如果在向量丛上给出一个通常意义上的芬斯勒度量，那么它在全空间的垂直子丛上导出一个黎曼内积，这样，芬斯勒几何就转化为这个黎曼向量丛的几何。人们很自然地会质疑我们为什么需要芬斯勒几何。为了回答这个问题，我们将描述复芬斯勒几何在某些不可能通过厄米几何研究的学科中的一些应用。
A Finsler metric of a manifold or vector bundle is defined as a smooth assignment for each base point a norm on each fibre space, and thus the class of Finsler metrics contains Riemannian metrics as a special sub-class. For this reason, Finsler geometry is usually treated as a generalization of Riemannian geometry. In fact, there are many contributions to Finsler geometry which contain Riemannian geometry as a special case (see e.g., [Bao et al. 2000], [Matsumoto 1986], and references therein). On the other hand, we can treat Finsler geometry as a special case of Riemannian geometry in the sense that Finsler geometry may be developed as differential geometry of fibred manifolds (e.g., [Aikou 2002]). In fact, if a Finsler metric in the usual sense is given on a vector bundle, then it induces a Riemannian inner product on the vertical subbundle of the total space, and thus, Finsler geometry is translated to the geometry of this Riemannian vector bundle. It is natural to question why we need Finsler geometry at all. To answer this question, we shall describe a few applications of complex Finsler geometry to some subjects which are impossible to study via Hermitian geometry.