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I'm confused with these definitions. The metric of certain space and the second fundamental form seem to be the same object.

I don't know what else to say, this is a pretty straight forward question.

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The first fundamental form and the metric tensor are indeed one and the same. The second fundamental form is not. It's a sort of derivative of the metric tensor.

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Thank you for confirming this! I actually meant the first fundamental form :) –  Diego Feb 15 '11 at 8:08

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