How to Derive and Interpret the Tensor Components of h in General Relativity?

[fu11meta1]

A map h: = T.(M) ---> T.(M) is defined by h(X) = X + g(U,X)U where U ε T.(M) is a fixed vector with g(U,U) = -1.

i: Give an expression for the components h^i (sub) j (This is "h" with a superscript i and subscript j) of h regarded as a tensor type (1,1)

ii: Prove that h^2 = h. Interpret h geometrically.

So I've been playing around with this but I'm getting no where. I could use some guidance on where to really get started. I'm also VERY new to general relativity, so every step/hint/anything would be great

 

[Matterwave]

So, the first step is probably to figure out what the question is really asking. So we have this map $h$ which takes a vector $X$ and gives you the back the vector $h(X)=X+g(U,X)U$. Since this mapping is taking a vector into another vector, then we know that $h$ is a type (1,1) tensor which has components $h^i_{~~j}$. So we know then that in component notation $h(X)\equiv h^i_{~~j}X^j e_{(i)}$ where $e_{(i)}$ are the basis vectors (the parenthesis mean that they are not components of a vector, but each $i$ denotes a different vector). Can you perhaps turn the right hand side of the first equation into component notation to see what it looks like?

 

[Orodruin]

Regarding ii, this should be a simple matter of insertion and using the linear property of g. For the interpretation, what happens if you set X = U? What happens when you use X such that g(U,X) = 0? (I suggest not using the coordinate representation for ii as it really is not necessary.)