Jon Herman, Pure Mathematics Department, University of Waterloo
“The second fundamental form”
When an immersed submanifold of a Riemannian manifold is endowed with the induced metric it is called a Riemannian submanifold. The second fundamental form gives a way to compare the structure of a Riemannian manifold to a Riemannian submanifold. After introducing the second fundamental form, I will show how it gives a relationship between the Riemannian connections (via the Gauss formula), curvatures (via the Gauss equation) and geodesic curvatures (via the Gauss formula along a curve) of a Riemannian manifold and its Riemannian submanifolds.