没有齐性和对称性假设的黎曼几何
Riemannian geometry without the hypotheses of homogeneity and symmetry
摘要：
一个广义的黎曼几何被认为是完全基于最小的假设，即线元素$ds$是一个经常性的功能的位置和方向，每一个点的距离本身是等于零。除了黎曼线元，也黎曼的剩余假设的均匀性和对称性下降。令人惊讶的是，无穷小的毕达哥拉斯距离公式重新出现，而不需要假设，作为$almost$ $every$几何的第一近似，它对于方向反转是不变的。更一般地说，$almost$ $every$几何的第一近似是一个单参数齐次黎曼-兰德斯线元族，自然地为经典电磁场和引力场的统一理论提供了几何框架。几何自然地解释了电磁和引力相互作用之间的层次结构、它们不同的吸引/排斥性质、电荷、CPT对称性、麦克斯韦和爱因斯坦方程。在这个框架内，高阶项可以描述可能与大尺度相关的新时空自由度。
英文摘要：
A generalisation of Riemannian geometry is considered, based exclusively on the minimal assumptions that the line element $ds$ is a regular function of position and direction and that the distance of every point from itself is equal to zero. Besides the Riemannian line element, also Riemann's residual hypotheses of homogeneity and symmetry are dropped. Surprisingly, the infinitesimal Pythagorean distance formula reemerges, without the need of being postulated, as a first approximation to $almost$ $every$ geometry that is invariant with respect to direction reversal. More in general, the first approximation to $almost$ $every$ geometry is a one-parameter family of homogeneous Riemann-Randers line elements, naturally providing the geometrical framework for a unified theory of the classical electromagnetic and gravitational fileds. Geometry naturally accounts for the hierarchy between electromagnetic and gravitational interactions, their different attractive/repulsive nature, electric charge, CPT symmetry, Maxwell and Einstein equations. Within this framework higher order terms could describe new spacetime degrees of freedom of possible large-scale relevance.
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