Let $M$ be a Riemannian manifold. What is the definition of a $C^{1,\alpha}$ ($0<\alpha<1$) Riemannian metric on $M$?
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Check this out. Being not sure about the context, I do not put it as an answer. – Gibbs Nov 14 '18 at 09:46
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I figured it out. Since $Sym^2(T^M)$ is a parallel subbundle of $T^M\otimes T^M$, it inherits a metric $h$ from $T^M\otimes T^M$, which is the canonical metric $g$ inherited by the Riemannian metric on $M$. So a $C^{1,\alpha}$ metric $g$ on $M$ is a $C^1$ map from $M$ to $Sym^2(T^M)$ with the covariant derivative $\nabla_i g$ is Holder continuous with exponent $\alpha$ and is positive definite at every $p\in M$ – Truong Nov 14 '18 at 13:01
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Good, so it was Hölder condition. – Gibbs Nov 14 '18 at 13:03
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The key point here is that one must find a canonical metric on $Sym^2(T^*M)$. – Truong Nov 14 '18 at 13:07