Ricci Flow on Modified Riemann Extensions

Abstract

The Ricci flow and the evolution equations of the Riemannian curvature tensor were initially introduced by Hamilton [8] and was later studied to a large extent by Perelman [13–15], Cao and Zhu [4], Morgan and Tian [10]. Indeed, the theory of Ricci flow has been used to prove the geometrization and Poincare conjectures [1]. However not much work has been done on Ricci flows on modified Riemann extensions. The Ricci flow equation is the evolution equation ∂gij ∂t = −2Rij where gij and Rij are metric components respectively. As flow progresses the metric changes and hence the properties related to it. Patterson and Walker [11] have defined Riemann extensions and showed how a Riemannian structure can be given to the 2n dimensional tangent bundle of an n-dimensional manifold with given non-Riemannian structure. This shows that Riemann extension provides a solution of the general problem of embedding a manifold M carrying a given structure in a manifold M carrying another structure, the embedding being carried out in such a way that the structure on M induces in a natural way the given structure on M . The Riemann extension of Riemannian or non-Riemannian spaces can be constructed with the help of the Christoffel coefficients Γijk of corresponding Riemann space or with connection coefficients Π i jk in the case of the space of affine connection [5]. The theory of Riemann extensions has been extensively studied by Afifi [1]. Though the Riemann extensions itself is rich in geometry, here in our discussions, the modified Riemann extensions fit naturally in to the frame work. Modified Riemann extensions are introduced in [2] and their properties we list briefly in the next section.