A generic result on Weyl tensor

Anna Maria Micheletti, Angela Pistoia

DOI: http://dx.doi.org/10.12775/TMNA.2018.046


Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|> 0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.


Weyl tensor; Yamabe problem; generic result

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