# Indices in differential geometry

Often times in differential geometry it is convenient to use Einstein summation notation, and there it is presented to beginning graduates and advanced undergraduates alike that if you see two indices that are the same letter with one upper and the other lower, written next to each other, then there is an implied summation symbol.

No seriously mathematically rigorous notion can depend on the particular way in which we choose to communicate, and so even though you could rip apart the Einstein summation convention, asking all sorts of fringe case questions of the type "does Einstein summation notation apply in this case" it is generally understood enough that it is just accepted as a way to communicate, no more or less treacherous than the fact that we write symbols on paper. (Or that in general we do not write out all the quantifiers or use logical symbols in prose.) It is in the same light that I ask the following questions, but I hope that any answerers will also alert me if there is rigorous content in any of this:

Sometimes people seem to derive more meaning than just a bookkeeping device for Einstein Summation out of whether an index is written in a particular position. I don't understand what people are trying to communicate, but I can reconstruct enough of it that you can hopefully help me:

1. Sometimes writing upper indices indicates something of a "dual" nature. For instance, Einstein summation is often used when we contract a dual vector applied to a vector.
2. Sometimes writing something upper indicates inversion. For instance, $g^{ij}$ is the $ij_{th}$ entry of the inverse matrix $(g_{ij})$ which is the matrix of the Riemannian metric from a particular coordinate basis to itself.
3. Sometimes the left to right order of the indices on a complicated tensor matters.
4. I believe I've seen indices written to the left of a mathematical symbol.

My questions are:

a. Is there any system to what the location of an index means? Or is it just special case by special case? For instance, is 2. an instance of a more general rule, or is it that inversion is only meant in the case of $g_{ij}$. 2. and 1. seem to be in conflict? Or maybe neither 1. nor 2. hold, there is no meaning, and it just happens that for both the case of inversion and transpose, it is convenient to simply write upper indices so that Einstein summation is in effect?

b. Which of these imply more significance than simply a bookkeeping device for Einstein summation notation, among points 1 and 2. above?

c. In points 3. and 4. I am not even aware of what the "face-value" meaning is, as in I would not know how to perform a computation with these things. For instance, is a left-side index supposed to mean transpose or something?

Feel free to simply connect me to a resource, but the most pedestrian searches on my part haven't revealed anything authoritative. I was unable to rigorously understand introductory differential geometry for most of my undergrad because few people stopped to explain their notation, so I hope that in addition to helping me right now, this might become useful to someone else struggling in the same way in the future.

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1. Sometimes writing upper indices indicates something of a "dual" nature. For instance, Einstein summation is often used when we contract a dual vector applied to a vector.

This is the only meaning of upper indices that I am aware of. That is, subscripts denote covariant components of a tensor, while superscripts denote contravariant components. See this Wikipedia section.

2. Sometimes writing something upper indicates inversion. For instance, $g^{ij}$ is the $ij$th entry of the inverse matrix $(g_{ij})$ which is the matrix of the Riemannian metric from a particular coordinate basis to itself.

This is a special case of #1. In particular, $g^{ij}$ represents the $ij$th component of the contravariant metric tensor. As a matrix, the contravariant metric tensor happens to be the inverse matrix of the covariant metric tensor $(g_{ij})$. See the Wikipedia article on raising and lowering indices.

3. Sometimes the left to right order of the indices on a complicated tensor matters.

In principle, the left to right order of the indices always matters. Of course, if a tensor happens to be symmetric, then the indices can be swapped without affecting the value.

4. I believe I've seen indices written to the left of a mathematical symbol.

I am not aware of this notation.

Which of these imply more significance than simply a bookkeeping device for Einstein summation notation, among points 1 and 2. above?

Perhaps the abtract index notation article will shed some light on this for you. Index notation is really fundamental to the nature of tensors.

Edit: Personally, I also feel like I gained a lot of insight into Einstein index notation when I learned about Penrose graphical notation. See the Wikipedia article, or The Road to Reality by Roger Penrose.

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You should try very simple examples just to get used to the notation:

Let $V$ be a vector space, $(e_1,\dots,e_n)$ a basis, $(e^1,\dots,e^n)$ a basis for the dual space $V^*$ and let $v\in V$, $h\in V^*$. Then $$v=v^i e_i$$ for a bunch of numbers $v^i$, and $$h=h_j e^j$$ for a bunch of numbers $h_j$. If you evaluate $h(v)$ you get $$h(v)=h_j e^j (v^i e_i) = h_j v^i e^j(e_i)=h_jv^j$$ which is kind of automatic. So vector-components have up-indices, dual vectors (aka forms) have there indices down. Now let $A$ be a homomorphism $V\rightarrow W$ and let $u=(u_1,\dots,u_m)$ be a basis of W. Then $$A(v)=A(v^i e_i)= v^i A(e_i) = v^i A^j_i u_j = A^j_i v^i u_j$$ for $nm$ numbers $A^i_j$, which we call a tensor of type 1,1.

Now $g_{ij}$ can take two vectors to produce a number ($g_{ij}v^iw^j$), while $g^{ij}$ takes two dual vectors, so you should really think of $g_{ij}$ as the matrix of a bilinear form.

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I believe I've seen indices written to the left of a mathematical symbol.

• One notation for binomial coefficients and related functions: ${}_nC_r = \binom{n}{r}$
• Hypergeometric functions: e.g. ${}_2F_1(a,b;c;z)$
• The spectral sequences associated to a filtration: e.g. ${}^{II} E^n_{p,q}$