# differential geometry : basic query about tensor notation and tensor products

I have a few very basic queries. I've been studying differential geometry as part of a course on General Relativity, so I don't have a very well grounded understanding of the mathematical formalism; it's all kind of a blurry mess of index notations. The question are:

1) How do I formally interpret the product of two tensors/vectors? We often write products of the type $v^iw^j$ and just interpret them as a rank 2 tensor. Is this expression commutative? i.e. is it the same as $w^jv^i$? I realize that this creature transforms like a rank 2 contra-variant tensor, but does it mean anything geometrically? Is there a meaningful notion of a product here? The same can be done for higher rank tensors, even products of mixed higher with lower rank tensors, etc.

2) Regarding the operation of lowering and raising indexes using the metric tensor. I realize that we define the co-variant vector $v_i$ as $g^{ij}v^j$. Can I write this as $v^jg^{ij}$? This kind of relates to the previous question regarding commutation. Do I interpret the raising/lowering operation as a linear operator operating on a vector? If so the second notation seems wrong to me, but I've seen it used before...

3) We often encounter equations like the following one for a Christoffel symbol - $\Gamma_{\,\mu\nu}^{\rho}=g^{\rho\rho'}\Gamma_{\rho,\mu\nu}$. I have a problem interpreting this type of equation, as for it to make sense, I generally expect to see the same indexes on both sides. However on the right side $\rho$ is a summation index and doesn't actually appear there. I realize this is "just" raising the index, but when I look at what's actually written it seems like abuse of notation. Am I misunderstanding this?

Thanks a lot in advance for any help

Index notation can for the most part be understood using just the Einstein summation convention - you're just dealing with scalar equations between the components of various vectors, matrices etc. (This is not quite true when it comes to e.g. covariant derivatives in abstract index notation, but is probably the easiest way to understand things to start with and makes computation very natural.)

Thus the tensor product $v \otimes w$ (represented in index notation by $v^i w^j$) is a matrix whose $(i,j)^\text{th}$ component is the multiplication $v^i w^j$, so $v^i w^j = w^j v^i$ by commutativity of scalar multiplication. This does not mean the tensor product is commutative, which would be $v^i w^j = w^i v^j$, i.e. symmetry of the matrix $v \otimes w$, which is not true in general.

Your raising/lowering expressions are incorrect - pairs of indices should always be one upper, one lower. The correct expression is $v_i = g_{ij} v^j = v^j g_{ij}$ - once again the order doesn't matter in this notation. You can indeed interpret $g$ as a linear operator - in this context it is a linear map $T_p M \to T_p M ^*$; i.e. from the contravariant tangent vectors to the covariant tangent vectors. I agree that $g_{ij}v^j$ is the more intuitive ordering, but either is correct.

I'm guessing your Christoffel expression is meant to be $\Gamma_{\,\mu\nu}^{\rho}=g^{\rho\rho'}\Gamma_{\rho'\mu\nu}$. This looks fine to me: $\rho$ is a free index on both sides, it's $\rho'$ that is summed over. They are completely different indices, not sure why the author decided $\rho'$ was appropriate.

If you really want to understand what tensors are, I recommend learning about tensors from the mathematical perspective - if you have background in linear algebra this shouldn't be too difficult. I recommend Chapter 2 of O'Neill's Semi-Riemannian Geometry with Applications to Relativity, but it may be difficult to approach depending on how rigorously you've been treating manifolds - perhaps someone can recommend a simpler introduction to multilinear algebra (since once you understand tensors at a point tensor fields should be very intuitive).

The short of it is that tensors are all multilinear maps, and all these operations that look like multiplication and summation in index notation are the natural ways to combine these maps. For example, consider two covectors $v_i$ and $w_i$ - these are linear maps that send vectors $x$ to scalars $v_i x^i,w_i x^i$. The tensor product is then the natural product of these - the bilinear map that sends two vectors $x,y$ to the scalar $v_i w_j x^i y^j$. Since $x,y$ are the inputs here, the bilinear map itself is $v_i w_j$.

• Thanks a lot for taking the time to write this out! It helped and cleared some stuff up. I did make a typo on the lowering index operation, sorry. Regarding my last question, what you write is what I would generally expect to see, what I wrote is what my professor consistently uses, so it's not a one time mistake; I guess it is kind of abuse of notation on his part? – user126743 Jan 31 '15 at 16:32
• Maybe I can find an example of this type of notation in some book. Anyways thanks a lot for the help! – user126743 Jan 31 '15 at 16:41
• @user126743: I guess it's a personal quirk - I'd ask him. – Anthony Carapetis Feb 1 '15 at 0:15