This is the famous theorem of Gromoll and Meyer:
Theorem (Gromoll-Meyer, 1974) There is an exotic 7-sphere with nonnegative
sectional curvature and positive sectional curvature at a point.
I don't understand the second part of theorem "positive sectional curvature at a point". Isn't it always possible that one cane perturb the metric such that it has positive sectional curvature at a point and we have $\sec_{\min}(M)\leq \sec_q\leq \sec_p$ for all $q$ in a small neighborhoods of $p$?
Best Answer
The short answer is that you cannot necessarily perturb a metric to get positive sectional curvature while still maintaining non-negative/positive sectional curvature everywhere else.
My favorite example of this is due to Wilking
He constructs a smooth Riemannian metric $g$ on $M:=\mathbb{R}P^2\times \mathbb{R}P^3$ with the following properties.
1) It is non-negatively curved everywhere.
2) If $U\subseteq M$ is the subset of all points $p$ for which every $2$-plane $\sigma \subseteq T_p M$ is positively curved, then $U$ has full measure in $M$
In other words, in the measure-theoretic sense, $M$ is positively curved almost everywhere (and non-negatively curved everywhere).
Note, however, that $M$ is non-orientable. The classical Synge theorem says that in odd dimensions, a positively curved closed Riemannian manifold must be orientable. Thus, the metric $g$ cannot be deformed to being positively curved everywhere, despite the fact that it started off with positive curvature almost everywhere.
Edit Following Travis's suggestion, I'll move part of my comment below into the answer here.
Proposition. Suppose $(M,g)$ is a Riemannian manifold and $p\in M$. Let $p\in U\subseteq M$ be an open set diffeomorphic to a ball in $\mathbb{R}^n$ Then $g$ can be deformed to a metric $g_1$ for which all sectional curvatures are positive near $p$, but $g= g_1$ outside of $U$.
Proof: Because $U$ is diffeomorphic to a ball, there is a diffeomorphism $f:U\rightarrow D^n_+\subseteq S^n$, where $D^n_+$ denotes the open northern hemisphere. Let $g_0$ denote the canonical (positively curved) metric on $S^n$.
Choose an open set $V\subseteq U$ with $\overline{V}\subseteq U$ and let $\lambda:M\rightarrow \mathbb{R}$ be a bump function with $\lambda \equiv 1$ near $p$, and $\operatorname{supp} \lambda \subseteq \overline{V}$.
The family of metrics $(1-t)g + \lambda t f^\ast g_0$ has the required properties.