Finding The Second Fundamental Form

Question

Using the following $$L_{i}^{ j}=-L_{ik}g^{kj}$$ we can calculate the second fundamental form.

But in exercise is see that from matrix multiplication it is $$L_{i}^{ j}=-g^{kj}L_{ik}$$ does it commute?

In the case of $\mathbb{R}^3$

Answer 1

I take it that $L_{ij}$ are the components of the second fundamental form and that therefore $L : T_{p}M \mapsto T_{p}M$ and $\mathcal{L} : T_{p}M \times T_{p} M \mapsto \mathbb{R}$? If so then $L_{ij} = \mathcal{L}(x_i , x_j )$. What you have then aritten above is the standard way of raisign and lowering the indeices using the metric and the correct way to write it is $L_{ij} = \sum_{k} g_{ik}L^{k}_{j}$.