Metrics Induced by \(\mathbb {R}^{6+k}\) on the Graph of k-Smooth Functions and the Hopf Conjecture

In this work, we first give a formula for the sectional curvature of a metric induced by \(\mathbb {R}^{6+k}\) on the graph of k-smooth functions on \(\mathbb {S}^2 \times \mathbb {S}^2\) . Secondly, we give a new proof of a theorem on metrics induced by \(\mathbb {R}^{7}\) on the graph of 1-smooth function. Finally, we show that a metric induced by \(\mathbb {R}^{8}\) on the graph of two smooth functions which are lifts on \(\mathbb {S}^2 \times \mathbb {S}^2\) does not have a positive sectional curvature.