Tuesday Geometry Working seminar

“Geometric Interpretations of Curvature”

Last time we discussed the concrete geometric meaning of Gaussian curvature of surfaces in R3. In this talk we will use the Gaussian curvature to give geometric meaning to the sectional, Ricci and scalar curvatures of an arbitrary Riemannian manifold. To do this, we will also need to define totally geodesic submanifolds. In the case of R3, it turns out that the first and second fundamental form completely classify the surfaces. This is made precise by a theorem of Bonnet, which I will state. After doing this, we will discuss the Codazzi equations and see how they, together with the Gauss equation, are generalizations of this classification theorem for surfaces.

Please note room