Normal Coordinates in Affine Geometry

Abstract

Manifolds endowed with an affine geometry of general type with nontrivial metric, torsion, and nonmetricity tensor are considered. Such manifolds have recently attracted much attention due to the construction of generalized gravity models. Under the assumption that all geometric objects are real analytic functions, normal coordinates in a neighborhood of an arbitrary point are constructed by expanding the connection and the metric in Taylor series. It is shown that the normal coordinates are a generalization of a Cartesian coordinate system in Euclidean space to the case of manifolds with any affine geometry. Moreover, the components of any real analytic tensor field in a neighborhood of any given point are represented in the form of a power series whose coefficients are constructed from the covariant derivatives and the curvature and torsion tensors evaluated at this point. For constant curvature spaces, these series are explicitly summed, and an expression for the metric in normal coordinates is found. It is shown that normal coordinates determine a smooth surjective mapping of Euclidean space to a constant curvature manifold. The equations for extremals are explicitly integrated in normal coordinates for constant curvature spaces. A relationship between normal coordinates and the exponential mapping is analyzed.