A riemannian metric is in a neighborhood of $g$ (?)

Question

What do we mean by this:

"A Riemannian metric $g_1$ is in a small $C^{l+1,\alpha}$ neighborhood of the metric $g$" ?

Any help is appreciated!

Answer 1

The notation $C^{l+1,\alpha}$ refers to a Hölder space. This is the space of functions with $l+1$ derivatives such that the $(l+1)^{\textrm{st}}$ derivative is Hölder continuous with exponent $\alpha$.

The statement that $g_1$ is in a $C^{l+1,\alpha}$ neighborhood of $g$ is a statement that, for $g_1$, the first $l+1$ derivatives (including the $0^\mathrm{th}$, i.e. the value of $g_1$ itself) and the Hölder coefficients (see the link above) of its $(l+1)^{\textrm{st}}$ derivative are close to those of $g$.