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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)?

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# Shape Operator for Schwarzschild spacetime in 2-dim

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