Mean Gaussian curvature from non-unit vector Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak.

Let us suppose that $\lambda_{h|}^i$
is not a unit vector and therefore, the mean curvature $M_h$ in
this case is given by
$$M_h=-\frac{R_{ij}\lambda_{h|}^i\lambda_{h|}^j}{g_{ij}\lambda^i_{h|}\lambda^j_{h|}} \tag{1}$$

A proof was not given so I am trying to derive this.
For when $\lambda_{h|}^i$ are components of an orthornormal set of vectors
$$M_h=\sum_{k}K_{hk}=\sum_k\frac{\lambda_{h|}^l\lambda_{k|}^i\lambda_{h|}^j\lambda_{k|}^rR_{lijr}}{\lambda_{h|}^l\lambda_{k|}^i\lambda_{h|}^j\lambda_{k|}^r(g_{lj}g_{ir}-g_{lr}g_{ij})}=\sum_k\frac{\lambda_{h|}^l\lambda_{k|}^i\lambda_{h|}^j\lambda_{k|}^rR_{lijr}}{(1)\cdot(1)-0}$$
$$=\lambda_{h|}^l\lambda^j_{h|}R_{lijr}g^{jr}=-\lambda_{h|}^l\lambda^j_{h|}R_{lj}$$
where $\sum_k\lambda^i_{k|}\lambda^r_{k|}=g^{ir}$.
Now for when $\lambda_{h|}^i$ are not components of an orthonormal set of vectors, succumbing to the same approach gives
$$M_h=\sum_k K_{hk}=\frac{\lambda_{h|}^l\lambda_{h|}^j R_{lijr}}{(g_{lj}\lambda_{h|}^l\lambda_{h|}^j)g_{pq}}\sum_k\frac{\lambda_{k|}^i\lambda_{k|}^r}{\lambda_{k|}^p\lambda_{k|}^q} \tag{2}$$
It is not obvious to me how $(2) \to (1)$. Modifying the definition of $M_h$
$$M_h=\frac{\sum_k\lambda_{h|}^l\lambda_{k|}^i\lambda_{h|}^j\lambda_{k|}^rR_{lijr}}{(g_{lj}\lambda_{h|}^l\lambda_{h|}^j)\sum_k g_{pq}\lambda^p_{k|}\lambda_{k|}^q}=-\frac{R_{ij}\lambda_{h|}^i\lambda_{h|}^j}{g_{ij}\lambda^i_{h|}\lambda^j_{h|}N} \tag{3}$$
which is the nearest I can get to $(1)$, but with an extra factor $\frac{1}{N}$.

What's gone wrong? Is there a way to derive $(1)$ or is $(1)$ a definition itself?
 A: It's really quite simple to obtain your equation (1) for a non-unit vector from the unit-vector result (your second, unnumbered equation).
Start from your second equation which gives the mean curvature (Ricci curvature) in terms of the Ricci tensor $\mathbf{R}$ and a unit vector ${\lambda}_{h|}$:
$$M_h=-\sum_{i,j}\lambda_{h|}^i\lambda^j_{h|}R_{ij}.$$
Keep in mind that the index $h$ labeling the direction of the vector is fixed, only the component indices $i,j$ are summed over. Now define a new vector $\tilde{\lambda}_{h|}$ in the same direction $h$, but with a non-unit length, so the unit vector ${\lambda}_{h|}$ is obtained by dividing $\tilde{\lambda}_{h|}$ by its length, obtained from the metric tensor $\mathbf{g}$:
$${\lambda}_{h|}=\frac{\tilde{\lambda}_{h|}}{\sqrt{\sum_{k,l}g_{kl}\lambda_{h|}^k\lambda_{h|}^l}}.$$
Combination of these two equations gives the desired expression,
$$M_h=-\frac{\sum_{i,j}\tilde{\lambda}_{h|}^i\tilde{\lambda}^j_{h|}R_{ij}}{\sum_{k,l}g_{kl}\tilde{\lambda}_{h|}^{k}\tilde{\lambda}_{h|}^{l}}.$$

Notice that there is no need to consider a non-orthogonal set, it's only the length of the single vector $\lambda_{h|}$ that is changed. So in your equation (2) you can still assume that the vectors $\lambda_{k|}$ with $k\neq h$ form an orthonormal set, that seems to be the origin of the difficulty you encountered.
