Previous |  Up |  Next


affine connection; Ricci tensor; Cauchy-Kowalevski Theorem
The question how many real analytic equiaffine connections with arbitrary torsion exist locally on a smooth manifold $M$ of dimension $n$ is studied. The families of general equiaffine connections and with skew-symmetric Ricci tensor, or with symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of $n$ variables.
[1] Dušek, Z., Kowalski, O.: How many are torsion-less affine connections in general dimension. to appear in Adv. Geom.
[2] Dušek, Z., Kowalski, O.: How many are affine connections with torsion. Arch. Math. (Brno) 50 (2014), 257–264. DOI 10.5817/AM2014-5-257 | MR 3303775 | Zbl 1340.53021
[3] Egorov, Yu.V., Shubin, M.A.: Foundations of the Classical Theory of Partial Differential Equations. Springer-Verlag, Berlin, 1998. MR 1657445 | Zbl 0895.35003
[4] Eisenhart, L.P.: Fields of parallel vectors in a Riemannian geometry. Trans. Amer. Math. Soc. 27 (4) (1925), 563–573. DOI 10.1090/S0002-9947-1925-1501329-4 | MR 1501329
[5] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Amer. Math. Soc., 1978. MR 1834454 | Zbl 0451.53038
[6] Kobayashi, S., Nomizu, N.: Foundations of differential geometry I. Wiley Classics Library, 1996.
[7] Kowalevsky, S.: Zur Theorie der partiellen Differentialgleichung. J. Reine Angew. Math. 80 (1875), 1–32.
[8] Kowalski, O., Sekizawa, M.: Diagonalization of three-dimensional pseudo-Riemannian metrics. J. Geom. Phys. 74 (2013), 251–255. DOI 10.1016/j.geomphys.2013.08.010 | MR 3118584 | Zbl 1280.53020
[9] Mikeš, J., Vanžurová, A., Hinterleitner, I.: Geodesic Mappings and some Generalizations. Palacky University, Olomouc, 2009. MR 2682926 | Zbl 1222.53002
[10] Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, 1994. MR 1311248 | Zbl 0834.53002
[11] Petrovsky, I.G.: Lectures on Partial Differential Equations. Dover Publications, Inc., New York, 1991. MR 1160355
Partner of
EuDML logo