What's the definition of $C^\alpha$ norm of a tensor?

Question

Recently I came across $C^{k,\alpha}$ convergence of metrics as well. I am confused how to define this norm and can't find a book on it. Is the harmonic coordinate a necessity? Can someone put a good reference?

Answer 1

For a definition of $C^{k,\alpha}$ as well as references see my answer here: Definition of Hölder Space on Manifold, where a function $f:M\to N$ between two $C^\infty$ manifolds (of finite or infinite dimension) being $C^{k,\alpha}$ is made sense: in local charts $f$ is $k$ times continuously differentiable and $D^kf$ satisfies a local $\alpha$-Hölder estimate on the local chart with target the space of (bounded) $k$-linear maps endowed with a natural norm.

To apply this to $C^{k,\alpha}$ metrics, let $M$ be a (finite dimensional) $C^\infty$ manifold. Then the total space $TM$ of the tangent bundle as well as the total space $\operatorname{Sym}^2(TM)$ of the bundle of symmetric $2$ tensors on $M$ have unique $C^\infty$ manifold structures with respect to which $TM\to M$ and $\operatorname{Sym}^2(TM)\to M$ are $C^\infty$ vector bundles. Let us also define the cone bundle $\operatorname{Sym}^2_+(TM)\to M$ of positive definite symmetric $2$-tensors on $M$; note that sections of $\operatorname{Sym}^2_+(TM)\to M$ are precisely Riemannian metrics on $M$.

Define a total ordering on $\mathbb{Z}_{\geq0}\times]0,1]$ by

$$(k,\alpha)< (l,\beta) \iff [k=l \text{ and } \alpha<\beta] \text{ xor } [k<l].$$

Define $\operatorname{Met}^{(l,\beta)}(M)=C^{(l,\beta)}(\operatorname{Sym}^2_+(TM)\to M)$ to be the space of $C^{(l,\beta)}$ Riemannian metrics on $M$, and for $(k,\alpha)\leq (l,\beta)$ and $\mathfrak{g}_\bullet:\mathbb{Z}_{\geq0}\cup\{\infty\}\to \operatorname{Met}^{(l,\beta)}(M)$ say $\lim_{n\to \infty}^{(k,\alpha)} \mathfrak{g}_n=\mathfrak{g}_\infty$ if on each trivializing chart for the cone bundle $\operatorname{Sym}^2_+(TM)\to M$, $\mathfrak{g}_n$ converges to $\mathfrak{g}_\infty$ in uniform $C^{(k,\alpha)}$ topology (e.g. as discussed in https://math.stackexchange.com/a/3892256/169085; in this case one needs to add the $\alpha$-Hölder seminorm for the $k$-th derivative to the formulas; see also Definition of semi-norms on $C^{k,r}(\mathbb{R}^n,\mathbb{R}^m)$).