| Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann's lecture (see [17]), he stated that "Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...". It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.;Keywords. Riemannian Geometry, metrical connections, Riemannian curvature, compact 2-manifold, Gauss-Bonnet Theorem. |
