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Let $H$ be hyperbolic n-space. Let $TH$ be the tangent bundle of $H$, endowed with its Sasaki metric. I have two questions:

  1. Is $TH$ isometric to $H$ times a flat n-space?
  2. What is the group of isometries of $TH$?
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    $\begingroup$ Search online for for master thesis "NATURAL METRICS ON TANGENT BUNDLES" by ELIAS KAPPOS. There is stated that if the tangent bundle with Sasaki metric has bounded sectional curvature, then it is flat. This answers 1 because the product of of a hyperbolic and Euclidean space has curvature in $[-1,0]$ but it also has a non-virtually abelian discrete isometry group, i.e. a hyperbolic lattice. $\endgroup$
    – Igor Belegradek
    Commented Dec 18, 2013 at 0:11
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    $\begingroup$ Reagarding 2, a natural guess is that the isometry group is $\mathrm{Iso}(H)$, which certainly acts isometrically. Fibers are flat. With these two facts in mind, the curvature formulas (see above master thesis) should provide enough information to answer 2. $\endgroup$
    – Igor Belegradek
    Commented Dec 18, 2013 at 0:15

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