1
$\begingroup$

hi,

i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ?

marco

$\endgroup$
15
  • 1
    $\begingroup$ Your question has answer no: en.wikipedia.org/wiki/Hitchin%E2%80%93Thorpe_inequality $\endgroup$ Feb 11, 2012 at 21:58
  • $\begingroup$ This might be an interesting question, but it comes out of nowhere for me. Is there some reason why you believe that this might be the case? Is there some particular construction or approach you have in mind for obtaining the Einstein metric? $\endgroup$
    – Deane Yang
    Feb 11, 2012 at 22:42
  • $\begingroup$ I suppose your question is ambiguous. When you refer to the tangent bundle, are you actually referring to the bundle as-stated, or are you referring to the total space of the bundle? $\endgroup$ Feb 11, 2012 at 23:05
  • $\begingroup$ Ryan, I assumed he was asking for an Einstein metric on the total space. $\endgroup$
    – Deane Yang
    Feb 11, 2012 at 23:09
  • $\begingroup$ actually i am refering to the following: if $M$ is a manifold as stated above (compact, riemannian , real analytic ...). Now consider the tangent bundle as a manifold $TM$ (as a new manifold). Does this manifold always carry a einstein metric? If no why exactly? $\endgroup$
    – william
    Feb 11, 2012 at 23:51

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.