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## Main Question or Discussion Point

Some define the extrinsic curvature tensor as

$$K_{\mu \nu} = h^{\ \ \ \sigma}_\nu h^{\ \ \ \lambda}_\nu \nabla_\sigma n_\lambda.$$

From the expression it seems like the index of the covariant derivative in can be any spacetime index. However, does it makes sence to ask what the covariant derivative of the normal vector n to a hypersurface, when n is only defined at the surface? How does one make sense of the change in n away from the hypersurface?

$$K_{\mu \nu} = h^{\ \ \ \sigma}_\nu h^{\ \ \ \lambda}_\nu \nabla_\sigma n_\lambda.$$

From the expression it seems like the index of the covariant derivative in can be any spacetime index. However, does it makes sence to ask what the covariant derivative of the normal vector n to a hypersurface, when n is only defined at the surface? How does one make sense of the change in n away from the hypersurface?