Normal Coordinates in Affine Geometry

Abstract

Manifolds of affine geometry of general type with nontrivial metric, torsion, and nonmetricity tensor have been considered. These manifolds have recently attracted much interest due to the construction of generalized gravity models. Assuming that all geometric objects are real analytic, normal coordinates have been constructed in the neighborhood of an arbitrary point by decomposing the connection and metric components to the Taylor series. It has been shown that normal coordinates generalize the Cartesian coordinates in the Euclidean space to the case of manifolds with affine geometry of general type. Components of an arbitrary real analytic tensor field in the neighborhood of each point can be expressed as power series with coefficients constructed from the covariant derivatives, curvature and torsion tensors computed at the decomposition point. The power series have been explicitly summed for constant curvature spaces, and an expression for the metric in normal coordinates has been found. It has been shown that normal coordinates define the smooth surjective map of the Euclidean spaces to constant curvature manifolds. Equations for extremals in the constant curvature spaces have been explicitly integrated in normal coordinates. The relation between normal coordinates and exponential map has been analyzed.

Keywords

• normal coordinates
• Gaussian coordinates
• Riemannian coordinates