# Is this mean curvature?

Suppose $N_t:=\partial B(p, t)\subset M^{n+1}$ be the distance sphere in a Riemannian manifold. Let $\{x_1, \cdots, x_n\}$ be a coordinate of the distance sphere $\partial B(p, t)$. Hence $\{x_1, \cdots, x_n, t\}$ is a coordinate of $M$. Denoted $g_{ij}=g(\partial/\partial x_i, \partial/\partial x_j)$ be the first fundamental form of $N_t$. Let $G=\sqrt{\det g_{ij}}$

My question is: what is the geometric meanning of $$tr(G^{-1}\frac{d}{dt}G)$$ I guess it is $2H$, where $H$ is the mean curvature of $N_t$ w.r.t. the outer normal $\partial/\partial t$. But I am not good at tensor calculations, I don't know how to transfer it into an orthonormal basis. (if G is diagonalized, then it seems inside the trace, it is sencond fundamental form, correct?)

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