# einstein metrics on the tangent bundle

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

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Your question has answer no: en.wikipedia.org/wiki/Hitchin%E2%80%93Thorpe_inequality – Ryan Budney Feb 11 '12 at 21:58
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? – Deane Yang Feb 11 '12 at 22:42
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? – Ryan Budney Feb 11 '12 at 23:05
Ryan, I assumed he was asking for an Einstein metric on the total space. – Deane Yang Feb 11 '12 at 23:09
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? – william Feb 11 '12 at 23:51