Definition of a tensor field Could anybody explain to me the following:

If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$
  where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative,
  then $T_{ij}$ is a tensor field.

I am new to all hese covector, covariant, etc. and am very confused.
I believe the definition of a tensor is one which transforms according to the tensor transformation rule... right? But I don't see how that applies here. 
Thank you in advance.

Here are two paths you can take toward solving the problem.


*

*You can take your definition of "tensor" as a set of indexed quantities which transform according to a certain rule.  You need to show that $T_{ij} = \nabla_iT_j - \nabla_jT_i$ transforms according to these rules.  You know the expression for the covariant derivative in terms of derivatives and Christoffel symbols, so expand out $\nabla_i T_j - \nabla_j T_i$ and simplify until you get a linear combination of quantities you already know are tensors.  Then you're done, since a linear combination of tensors is again a tensor.

*To work from a coordinate-free vantage point, you can use a slightly different definition of "tensor": a $(p,q)$-tensor is a pointwise linear map which takes as input $p$ vector fields and $q$ covector fields and returns a function.  From this perspective, you have defined the function $T(X,Y) = \nabla_XY - \nabla_YX$.  You would like to show it is a tensor.  So show that it is pointwise linear in both inputs, i.e., that for any smooth functions $f,g$, $T(fX,gY) = fgT(X,Y)$.
For more, you might check out the Wikipedia page on the torsion tensor.
