# A Fundamental definition of extrinsic curvature

#### PLuz

My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface?

Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an hypersurface orthogonal congruence of time-like curves has zero vorticity and the many definitions that I've seen assume this fact. My guess is that the fundamental definition should be:

$$K_{ab}=\frac{1}{2}\mathcal{L}_n~h_{ab}~,$$

where $h_{ab}$ represents the induced metric on the hypersurface and $\mathcal{L}_n$ the Lie derivative along the normal to the hypersurface. By the way, should there be a minus sign in the above expression, I have seen both cases and it should not be irrelevant?

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#### martinbn

There is no such thing as "the fundamental" definition of anything. Any two equivalent definitions are equally fundamental although one may be more convenient in some situations.

The definition of extrinsic curvature does not even mention vorticity or a congruence of curves.

#### PLuz

There is no such thing as "the fundamental" definition of anything. Any two equivalent definitions are equally fundamental although one may be more convenient in some situations.

The definition of extrinsic curvature does not even mention vorticity or a congruence of curves.
Indeed, a definition is a definition. Yet, definitions can be made broader, or more generic. Maybe fundamental was a bad choice of words.
In either case, I understand the extrinsic curvature as the tangential rate of change of the normal to an hypersurface. Now, the rate of change: is it computed using the covariant derivative, the Lie derivative? Just so happens that in the context that is usually studied in the literature, it does not matter, both will define the same quantity that, however, does not have to be the case, in general. So, there must be a more generic - fundamental - way of defining the extrinsic curvature. A practical example, I have seen defining the extrinsic curvature of an hypersurface as $K_{ab}=h_a^i h_b^j \nabla_{(i}n_{j)}$ but in general this is not the same as the expression in the original post... Care to help me?

#### martinbn

I am not sure what you are looking for. The most general definition or the equivalence of the different definitions. Clearly something bothers you, but I am not sure what. It could be that you are not yet used to the new notion.

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