# Multiplication subscript

I'm teaching myself some differential geometry in the hope to understand gauge theory properly. In the definition of the the pullback bundle I came across a strange notation that I've never seen before. The pullback bundle was defined, with $$f: M \to N$$, as

$$\begin{equation} f^\ast \mathcal{E} \equiv (f^\ast E, f^\ast \pi, M , F), \end{equation}$$ where the total space is given by $$\begin{equation} f^\ast E \equiv N \times_M E := \{ (x,z) \in N \times E | f(x) = \pi(z) \}. \end{equation}$$

What does the subscript on the $$\times$$ operator mean? Is it something to do with the operation occurring on the manifold $$M$$?

• Are you familiar with Pullbacks in category theory? If not i think reading through this will help you understand the notation: en.wikipedia.org/wiki/Pullback_(category_theory) – Riquelme Jul 12 at 13:15
• Ah perfect, thanks! – SBrents Jul 12 at 14:55
• SBrents Can you please give a reference where you found it? – magma Jul 13 at 14:04

The idea is that you are given two functions both with the same target $$M$$, in this case $$f : N \to M$$ and $$\pi : E \to M$$. You want to form a kind of restricted product of $$N$$ and $$E$$, which could be called the "product of $$N$$ with $$E$$ over $$M$$". This is a subset of the "true" product, and it's defined by the formula that you gave.