| Abstract | In the field of Riemannian geometry, the condition on the Riemannian metric so that a manifold has positive scalar curvature (PSC) is important for a number of reasons. Many famous researchers have contributed gradually to this area of geometry, and in this project, we study more about PSC metrics on such manifolds. Specifically, we refine and provide some details to the proof of Gromov and Lawson that the connected sum of 2 n-dimensional manifolds will admit a PSC metric, provided each of the manifolds has a metric with the same condition. We then derive some useful formulas related to the Riemann curvature tensor, the Ricci tensor, and the scalar curvature in many different scenarios. We compute the quantities for a manifold equipped with an orthonormal frame and its dual coframe, namely the connection one-form and the curvature two-form. Then, we observe the change in the structure functions, defined as a function that determines the Lie derivative of the orthonormal frame, under a nearly conformal change of the said frame. The aim of these calculations is that, by expressing the scalar curvature of a manifold M entirely in terms of the structure functions, we can determine a condition on the conformal factor so that when dividing the tangent bundle of M into two sub-bundles, then the scalar curvature restricted to one sub-bundle will dominate that of the other one so that if we know the scalar curvature of the former sub-bundle is positive, we can be assured that the scalar curvature of M as a whole is also positive. |
