The extrinsic curvature $\mathcal{K}_{ij}$ of a sub-manifold (say, a boundary $\partial M$ of a manifold $M$) is defined as
$$\mathcal{K}_{ij} \equiv \frac{1}{2}\mathcal{L}_{n}h_{ij} = \nabla_{(i)}n_{j)},$$
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?