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Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution involution $σ$ in $Aut(G)$. Thus $σ$ is an automorphism of $G$ with $σ^2 = id$ and $H$ is an open subgroup of the set $G^\sigma=\{ g\in G: \sigma(g) = g\}.$

Now let $G/H$ be symmetric space then $T^*(G/H)\cong G^{\mathbb C}/H^{\mathbb C}$ is symmetric space?. Where $G^{\mathbb C}$ means complexification of lie group $G$

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    $\begingroup$ $T^*(S^2)$ is not diffeomorphic to any symmetric space. $\endgroup$ – Misha Apr 8 '14 at 19:24
  • $\begingroup$ thanks a lot for nice comment, but can you explain more why? $\endgroup$ – user21574 Apr 8 '14 at 19:25
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    $\begingroup$ de Rham decomposition: Every simply-connected symmetric space is isometric to a product of compact symmetric spaces, Euclidean spaces and symmetric spaces of noncompact type. In this example, the manifold does not split topologically as a product of $S^2$ and $R^2$ (why?). $\endgroup$ – Misha Apr 8 '14 at 21:45
  • $\begingroup$ Helgason in his book said that if $\frak g/\frak h$ is symmetric then $\frak g^{\mathbb C}/\frak h^{\mathbb C}$ is symmetric. So $G^{\mathbb C}/H^{\mathbb C}$ is symmetric , where is the point? $\endgroup$ – user21574 Apr 23 '14 at 16:32
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    $\begingroup$ My guess is that you are misreading what Helgason said. What you wrote is a vector space (quotient of two Lie algebras); of course, it admits a flat metric which is symmetric. This has nothing to do with your question. $\endgroup$ – Misha Apr 23 '14 at 17:46
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The answer is yes. See this paper http://arxiv.org/abs/0710.1543 (Introduction, and section 4.2 page 14). The basic ideas are that

  • if $M$ is a symmetric space, then $TM$ has a canonical structure of symmetric space (whatever the way you realize your symmetric space, apply the tangent functor to it);
  • the duality of vector bundles extends to a duality of "symmetric bundles" as defined in the paper (i.e., vector bundles in the category of symmetric spaces): the dual vector bundle of a symmetric bundle inherits a structure of symmetric space which makes it a symmetric bundle.

Hence, $T^*M$ inherits a canonical structure of symmetric space from that of $TM$.

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