# Covariant derivative of second fundamental form

Picture below is from 1706th page of Zhao Liang's The first eigenvalue of Laplace operator under powers of mean curvature flow.

$g_{ij}$ is Riemannian metric, $h_{ij}$ is second fundamental form, and $H=g^{ij}h_{ij}$. $\nabla$ is Riemannian connect, and $\nabla^pH=g^{pl}\nabla_lH$. $h$ is a constant independ to $x$.

For getting the below equation, I get stuck at $$g^{ij}g^{pl}H^k\nabla_lh_{ij}-2g^{ij}g^{pl}H^k\nabla_ih_{jl}=-H^k\nabla^pH$$ How should I to do it ?

Note

$$g^{ij} \nabla_l h_{ij} = \nabla_l (g^{ij} h_{ij}) = \nabla _l H$$

as $\nabla_l g^{ij} = 0$. Thus

$$g^{ij}g^{pl}H^k\nabla_lh_{ij}=g^{pl} H^k\nabla_l H = H^k\nabla^pH$$

by definition of $\nabla ^p = g^{pl}\nabla_l$.

For the next term, it is important to remark that one is working on hypersurfaces in $\mathbb R^{n+1}$. In particular the ambient curvature is zero and the Codazzi equation gives

$$\nabla_i h_{lj} - \nabla_l h_{ij} =0.$$

Thus we have $$-2g^{ij}g^{pl}H^k\nabla_ih_{jl}=-2g^{ij}g^{pl}H^k\nabla_lh_{ij}=-2g^{pl}H^k\nabla_lH = -2H^k\nabla^pH.$$