Let $M$ be a hypersurface of $\mathbb{R}^d$, $x_0 \in M$ and $T$ the tangent space of $M$ at $x_0$. Which of the following are true ?
If the sectional curvature of $M$ at $x_0$ is strictly positive then $M$ is locally of the same side of $T$ near $x_0$.
If the sectional curvature of $M$ at $x_0$ is strictly negative then $M$ is not locally of the same side of $T$ near $x_0$.
I know this statement is true if $M$ is a surface of $\mathbb{R}^3$ but I don't know for other manifolds.