Forgive me for asking a rather silly question, but I have thinking about the following definition of the extrinsic curvature ##\mathcal{K}_{ij}## of a sub-manifold (say, a boundary ##\partial M## of a manifold ##M##):(adsbygoogle = window.adsbygoogle || []).push({});

$$\mathcal{K}_{ij} \equiv \frac{1}{2}\mathcal{L}_{n}h_{ij} = \nabla_{(i}n_{j)},$$

where ##n## is the inward-pointing unit normal to ##\partial M## and ##h_{ij}## is the induced metric on ##\partial M##.

Is there an intuitive way to understand why the above must be the definition of the extrinsic curvature of a submanifold?

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# A Intution behind the definition of extrinsic curvature

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