$\require{AMScd}$ I'm reading a survey paper on closed geodesics, and this came up:
He constructs a bundle $Y_1$ over $T^1S^n$, where each fiber over $(x,v)$ is the $n-1$-dimensional sphere passing through $x$, orthogonal to the great circle which is tangent to $v$.
Now he proceeds to define $Y_k:= Y_1 \times _{ev}Y_1 \times _{ev} ... Y_1 $, where $ev:Y_1 \rightarrow S^n$ is, as he says, "the evaluation map at the origin composed with the representation map $\phi:Y_1 \rightarrow \Lambda S^n$ (the free loop space).
I thought the $\times _{ev}$ meant the pullback bundle given by the diagram:
\begin{CD} Y_1 \times_{ev} Y_1 @>>> Y_1\\ @V V V\ @VV ev V\\ Y_1 @>>ev> S^n \end{CD}
But this doesn't seem the case, since he later says that some maps define a fiber bundle structure on $Y_k$. My question therefore is: What is this product?