The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by Wald) prefer to do things entirely using tensor index notation (for example introduce Ricci & Riemann tensors). I do not always find it easy to translate between the two notations especially when exterior derivative and the totally antisymmetric levi-civita tensor are involved. Is there a good reference where I can learn how to translate between two notations ?
Also I feel that I am comfortable with using forms, but not so much with using tensor algebra & especially in the index notation. Is there a good place where geometry is done using tensors alone ? References for learning tensors independently of geometry will also be useful.
Best Answer
Generally speaking if you have a tensor $T$ on a manifold, and if you have a collection (of usually coordinate) vector fields $e_1, \cdots, e_n$ the "index notation" for $T$ is (lets assume for a moment $T$ is bilinear):
$$T_{ij} = T(e_i,e_j)$$
meaning $T_{ij}$ is a real-valued function for all $i$, and $j$. $T_{ij}$ is defined wherever the vector fields $\{ e_i : i = 1,2,\cdots n\}$ are defined. On a manifold with a metric (meaning an inner product on every tangent space), it is typical to define
$$g_{ij} = \langle e_i, e_j \rangle$$
where $\langle \cdot, \cdot \rangle$ is the inner product on the tangent spaces.
If the tensor takes something other than two vectors as input, for example the Riemann curvature tensor is sometimes thought of as a bilinear function from the tangent space to the space of skew-adjoint linear transformations of that tangent space, i.e. at every point $p$ of the manifold it is bilinear $T_p N \oplus T_p N \to Hom(T_p N, T_p N)$ taking values in the skew-adjoint maps (with respect to the inner product). So given $e_i, e_j \in T_p N$, $R(e_i,e_j)$ is a linear functional on the tangent space, so you could express $R(e_i,e_j)(e_k)$ as a linear combination of vectors in the dual space $T^*_p N$. The standard basis vectors of the dual space (corresponding to the collection $\{e_i\}$) is typically denoted $e_1^*, \cdots, e_n^*$. So you write $R(e_i,e_j)(e_k) = \sum_l R^l_{ijk}e^*_l$, and call $R^l_{ijk}$ the Riemann tensor "in coordinates".
In case any of this is unfamiliar, $e^*_j(e_i) = 1$ only when $i=j$ and $e^*_j(e_i) = 0$ otherwise. Or "in coordinates" $e^*_j(e_i) = \delta_{ij}$.
I think many intro general relativity textbooks explain this fairly well nowadays. When I was an undergraduate I liked: