Normal Coordinates in Affine Geometry

M.O. Katanaev

Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia

Kazan Federal University, Kazan, 420008 Russia

E-mail:  katanaev@mi.ras.ru

Received April 25, 2016

Full text PDF

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

References

1. Riemann B. Über die Hypothesen, welche der Geometrie zu Grunde liegen, Nachrichten von der Gesellschaft von Wissenschaften, Göttingen, 1868, vol. 13. pp. 133–152. (In German)

2. Eisenhart L.P. Riemannian Geometry. Princeton, Princeton Univ. Press, 1926, 252 p.

3. Cartan É. Leçons sur la géométrie des espaces de Riemann. Paris, Gauthier–Villars, 1928. 273 p. (In French)

4. Rashevskii P.K. Riemannian Geometry and Tensor Analysis. Moscow, Nauka, 1967. 664 p. (In Russian)

5. Petrov A.Z. New Methods in General Relativity Theory. Moscow, Nauka, 1966. 496 p. (In Russian)

6. Katanaev M.O. Geometrical Methods in Mathematical Physics. Applications in Quantum Mechanics. Part 1. Moscow, MIAN, 2015. 176 p. (In Russian)

7. Katanaev M.O. Geometrical Methods in Mathematical Physics. Applications in Quantum Mechanics. Part 2. Moscow, MIAN, 2015. 185 p. (In Russian)

8. Eisenhart L.P. Non-Riemannian Geometry. New York, Am. Math. Soc., 1927. 184 p.

9. Schouten J.A., Struik D.J. Einfuhrung in die neueren Methoden der Differntialgeometrie. B. I. Groningen, Noordhoff, 1935. 338 p. (In German)

10. Fermi E. Sopra i fenomeni che avvengono in vicinanza di una linea oraria. Atti Acad. Naz. Lincei Rend. Cl. Sci. Fiz. Mat. Nat., 1922, vol. 31, pp. 21–23, 51–52, 101–103. (In Italian)

11. Katanaev M.O. Killing vector fields and a homogeneous isotropic universe. Phys. Usp., 2016, vol. 59, no. 7, pp. 689–700. doi: 10.3367/UFNr2016.05.037808.

12. Katanaev M.O. Lorentz invariant vacuum solutions in general relativity. Proc. Steklov Inst. Math., vol. 290, no. 1, pp. 138–142. doi: 10.1134/S0081543815060127.

13. Whitehead J.H.C. Convex regions in the geometry of paths. Q. J. Math., Oxford Ser., 1932, vol. 3, pp. 33–42.

For citation: Katanaev M.O. Normal coordinates in affine geometry. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 47–63. (In Russian)

The content is available under the license Creative Commons Attribution 4.0 License.