I'm having a hard time getting a grasp of what $g^{a}{}_{b}$ truly means in GR. I understand that somehow it is the metric tensor in a sense, but I don't know its properties. I have some wild guesses about it, but I'm really not sure.
Is it just the metric tensor with one index pulled up by the inverse metric ($g^{a}{}_{b}=g^{a\mu}g_{\mu b}$)? If yes, what are its properties?
Is it Correct to say that it is a tensor of type (1,1) (since the one upper and one lower index)? If yes, how can I imagine that it is a linear map from a vector space to itself?
I couldn't answers to my questions on the internet, so thanks in advance for replying!
