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Let $X(s,t), (s,t)\in E$, with $E\subset \mathbb{R}^2$ a compact set, be acentered two dimensional Gaussian random field with continuous trajectories andvariance function $\sigma(s,t)$. Denote by $\mathcal{L}=\{(s,t):\sigma(s,t)=\max_{(s',t')\in E}\sigma(s',t')\}$. In this contribution, wederive the exact asymptotics of $\mathbb{P}\left(\sup_{(s,t)\inE}X(s,t)>u\right)$, as $u\to\infty$, under condition that $\mathcal{L}$ is asmooth curve. We illustrate our findings by an application concerned withextremes of the aggregation of two independent fractional Brownian motions.

Extremes of Gaussian Random Fields with maximum variance attained over smooth curves