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Let's start from the begin to see where I get lost.

Extrinsic curvature defines the way an object relates to the radius of curvature of circles that touch the object (a couple of further nicer definitions come from physics, for the moments I am not mentioning them), and intrinsic curvature defines each point in a Riemannian manifold.

A Mean curvature locally describes the curvature of an embedded surface.

[itex]1/2 (k_1 + k_2)[/itex] as extrinsic.

A Gaussian curvature is defined as an intrinsic curvature as [itex] k_1 \times k_2 [/itex], the reason of this is just that Gaussian curvature involves surfaces ? Then why cannot we define the mean curvature as well as intrinsic

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# Why the Gaussian curvature is extrinsic ?

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