I'm reading Petersen's Riemannian geometry. Here the Ricci curvature is defined as the trace of $R$, where $R$ is the curvature tensor. More precisely,if $e_1,...e_n \in T_p M $ is an orthonormal basis,then $$\begin{align} \mathrm{Ric}(v,w)&=\mathrm{tr}(x \rightarrow R(x,v)w) \tag1\\ &=\sum_{i=1}^n g(R(e_i,v)w,e_i) \tag2\\ \end{align}$$
Here is my confusion:
Why $(1)$ implies $(2)$? I've referred some books about tensor's trace or contraction but I cannot figure it out in a good way.
The same as the definition of Scalar curvature, which is the trace of Ricci curvature. More precisely: $$\begin{align} \mathrm{Scal}&=\mathrm{tr}(\mathrm{Ric}) \tag3\\ &=\sum_{i,j=1}^n g(R(e_i,e_j)e_j,e_i) \tag4\\ &=2\sum_{i\lt j} \mathrm{sec}(e_i,e_j) \tag5\\ \end{align}$$
Why $(3)$ can imply $(4)$ and $(5)$?
Also, if conveniently, can anyone else give me a good reference or notes for tensors? I'm not habituated to the language of tensors, thanks.
(1) implies (2) just by the definition of the trace of the linear operator $x\mapsto R(x,v)w$. In general we compute the trace of a linear operator $L$ in an inner product space $\langle \cdot, \cdot \rangle$ by taking any orthonormal basis $e_1,\ldots, e_n$ and then $\mathrm{tr}(L)=\sum_{i=1}^n \langle L(e_i), e_i\rangle$.
In your case, $\mathrm{Ric}(v,w)=\mathrm{tr}(x\mapsto R(x,v)w)=\sum_{i=1}^n g(R(e_i, v)w, e_i)$, which is (2).
The Ricci that appears in (3) is not the Ricci curvature. In fact, is the Ricci operator defined by $g(\mathrm{ric}(v),w)=\mathrm{Ric}(v,w)$. It is easy to verify that $\mathrm{ric} $ is linear so
$\mathrm{scal}=\mathrm{tr}(\mathrm{ric})=\sum_{j=1}^n g(\mathrm{ric}(e_j),e_j)=\sum_{j=1}^n \mathrm{Ric}(e_j,e_j)$ which is (because of (2)) equal to $\sum_{i,j=1}^n g(R(e_i,e_j)e_j,e_i)$.
Finally, verifying that the Sectional curvature is equal to (4) is just the definition I think.
A good reference to tensors if I'm not wrong is O Neill's book, "Semi-riemannian Geometry".
Another interesting Riemannian Geometry books apart from Petersen's book are Lee's book "Riemannian Manifolds", and Do Carmo's book "Riemannian geometry".