Hello: I would like to understand how to compute the shape operator (and eigenvalues etc) for a complex example like the Schwarzschild spacetime. It's easy for a submanifold in Euclidean space, but I don't know how to do it for the more advanced examples like the schwarzschild spacetime in 2-dim. QUESTION: Given the Schwarzschild metric in Eddington-Finkelstein coordinates in 2- dim: ds^2 = -(1 - 2M/r)dv^2 + 2dvdr , which is a surface. How can I compute the Schwarzschild a) Shape operator (Weingarten map or second fundamental tensor) b) unit normal vectorfield c) Eigenvalues of the Shape operator matrix? I know all the definitons etc., but the 2-dim Schwarzschild is not the typical surface/ submanifold embedded in Euclidean space. Could someone show me how to compute the steps a)-c) so that I understand it for this example (and can apply it to other examples/metrics)?