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Riemannian manifold

   Also found in: Wikipedia 0.01 sec.
Riemannian manifold [rē′män·ē·ən ′man·ə‚fōld]
(mathematics)
A differentiable manifold where the tangent vectors about each point have an inner product so defined as to allow a generalized study of distance and orthogonality.


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Further we have defined a semi-Einstein space, given an example and proved that in a Ricci-recurrent Riemannian manifold energy momentum tensor T is generalized recurrent if and only if the Einstein tensor G is generalized Ricci-recurrent and G will be generalized Ricci-recurrent if and only if the manifold is semi-Einstein.
Henry-Labordere (Societe Generale) then presents the heat kernel expansion on a Riemannian manifold, derives an asymptotic implied volatility in the context of local volatility models, and applies mathematical tools to the practical problems of pricing multi-asset options, interest rate models, and portfolio optimization.
He proved that in order that a Riemannian manifold admits a semi-symmetric metric connection whose curvature tensor vanishes, it is necessary and sufficient that the Riemannian manifold be conformally flat.
 
 
 
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