# Which tensor should the metric raising or lowering?

For something like $$g^{ij} n_i h_{kj}$$, how do I know which one should the metric operate on? $$n_i$$ or $$h_{kj}$$? The results could be $$n^j h_{kj}$$ or $$n_i h^i_{k}$$, which are different.

The question also appears when encountering partial derivatives too, like $$g_{ij}\partial_i h^i$$

• The results could be $n^j h_{kj}$ or $n_i h^i_{k}$, which are different. Why do you think those are different? Apr 11, 2021 at 17:39
• What I was caring is the notation. Do you mean both give one-form as results so they have the same physical meaning? Apr 11, 2021 at 17:42
• When you studied Special Relativity, did you understand that $a^\mu b_\mu$ and $a_\mu b^\mu$ are the same contraction $a\cdot b$? Apr 11, 2021 at 17:47
• I thought that case is different from this one cuz $g_{\mu \nu} a^\mu b^\nu$, no matter how I operate, it still gives me a vector and an one-form. While this case gives vector with (0 2) tensor and one-form with (1 1) tensor. Now I see summing up gives same h_k, thanks! Apr 11, 2021 at 17:59
• But what about the partial derivative case? For example, consider $\partial_i \partial_j h^i$, I could think of $\Box h^i$ and $\partial_i \partial_j h_j$. It seems to me that they are not the same. Apr 11, 2021 at 18:04