Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformaly equivalent to a product metric?

More generally, assume that $M$ and $N$ are two manifolds. What obstructions are existed for a

  metric $g$ on $M \times N$, to be conformally equivalent to a product metric for metrics

 $g_{1}$ and $g_{2}$ on $M$ and $N$, respectively?

Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformally equivalent to a product metric?

More generally, assume that $M$ and $N$ are two manifolds. What obstructions are there for a metric $g$ on $M \times N$ to be conformally equivalent to a product metric for metrics  $g_{1}$ and $g_{2}$ on $M$ and $N$, respectively?

Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformaly equivalent to a product metric?

More generally, assume that $M$ and $N$ are two manifolds. What obstructions are existed for a

metric $g$ on $M \times N$, to a be conformally equivalent to a product metric for metrics

$g_{1}$ and $g_{2}$ on $M$ and $N$, respectively?

Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformaly equivalent to a product metric?

More generally, assume that $M$ and $N$ are two manifolds. What obstructions are existed for a

metric $g$ on $M \times N$, to be conformally equivalent to a product metric for metrics

$g_{1}$ and $g_{2}$ on $M$ and $N$, respectively?