Revista de la
Unión Matemática Argentina
Special affine connections on symmetric spaces
Othmane Dani and Abdelhak Abouqateb

Volume 68, no. 1 (2025), pp. 327–342    

Published online (final version): June 13, 2025

https://doi.org/10.33044/revuma.4035

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Abstract

Let $(G,H,\sigma)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by $\nabla^0$ the canonical affine connection on the symmetric space $G/H$. A torsion-free $G$-invariant affine connection on $G/H$ is called special if it has the same curvature as $\nabla^0$. A special product on $\mathfrak{m}$ is a commutative, associative, and $\operatorname{Ad}(H)$-invariant product. We show that there is a one-to-one correspondence between the set of special affine connections on $G/H$ and the set of special products on $\mathfrak{m}$. We introduce a subclass of symmetric pairs, called strongly semi-simple, for which we prove that the canonical affine connection on $G/H$ is the only special affine connection, and we give many examples. We study a subclass of commutative, associative algebra which allows us to give examples of symmetric spaces with special affine connections. Finally, we compute the holonomy Lie algebra of special affine connections.

References

  1. S. Benayadi and M. Boucetta, Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures, Differential Geom. Appl. 36 (2014), 66–89.  DOI  MR  Zbl
  2. W. Bertram, The geometry of Jordan and Lie structures, Lecture Notes in Math. 1754, Springer, Berlin, 2000.  DOI  MR  Zbl
  3. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80, Academic Press, New York-London, 1978.  MR  Zbl
  4. N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ. 39, American Mathematical Society, Providence, RI, 1968.  MR  Zbl
  5. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience Publishers, New York-London, 1963.  MR  Zbl
  6. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, Interscience Publishers, New York-London, 1969.  MR  Zbl
  7. O. Loos, Symmetric spaces. I: General theory, W. A. Benjamin, New York-Amsterdam, 1969.  MR  Zbl
  8. A. Nijenhuis, On the holonomy groups of linear connections. IA, IB. General properties of affine connections, Indag. Math. (Proc.) 56 (1953), 233–240, 241–249.  MR  Zbl
  9. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 no. 1 (1954), 33–65.  DOI  MR  Zbl
  10. M. M. Postnikov, Geometry VI: Riemannian geometry, Encyclopaedia Math. Sci. 91, Springer, Berlin, 2001.  DOI  MR  Zbl