Let $0<\xi_1 <\xi_2<\ldots <\xi_n<1$ be $n$ variables, $\xi=(\xi_1,\ldots,\xi_n)$; and $\Gamma=\{\xi|0<\xi_1<\ldots <\xi_n<1\}$.
Let $U(\cdot,\cdot)\colon\Gamma\times [0,1]\to\mathbb{R}$ be a function for which we know that $U(\xi,\cdot)$ is a continuous function with a piecewise continuous first derivative.
Then, the map $\xi\to U(\xi,\cdot)$ defines an $n$-dimensional manifold $W\subset H^1$.
I think what is meant is that $W=\{U(\xi,\cdot):\xi\in\Gamma\}\subset H^1$ is an $n$-dimensional manifold. But I do not understand why.
Maybe someone could give me an explanation.
You must be missing some additional information. For example, if $U(\xi,\cdot)\equiv 0$, then it fulfills your assumptions, but $W=\{0\}$ is a single point, so cannot be an $n$-dimensional manifold.