on $M \times [0,T]$, where $\Delta_t$ denotes the rough Laplacian on $\operatorname{Sym}^2(T^*M)$ induced by $g(t)$, and where $N$ is a locally Lipschitz fiber-preserving bundle map on $\operatorname{Sym}^2(T^*M)$ depending continuously on $t$.
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Assume that for every $x \in M$, every $t \in [0,T]$, every nonnegative symmetric [bilinear form](/page/Bilinear%20Form)
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\begin{align*}
A \in \operatorname{Sym}^2(T_x^*M)
\end{align*}
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and every vector $v \in T_xM$ satisfying $A(v,v)=0$, one has
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\begin{align*}
N(A,t)(v,v) \ge 0.
\end{align*}
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If $h(x,0)$ is nonnegative as a symmetric bilinear form on $T_xM$ for every $x \in M$, then $h(x,t)$ is nonnegative as a symmetric bilinear form on $T_xM$ for every $x \in M$ and every $t \in [0,T]$.