Is sectional curvature constant if its constant on a special local orthonormal frame?

Question

Is sectional curvature constant if it constant on a special local orthonormal frame? In other word, If $(M,g)$ be a Riemannian manifold such that $K(e_i,e_j)=c$ then $K=$ constant at every point? where $\{e_1,\cdots,e_n\}$ is a special local orthonormal frame.

Answer 1

No, it's not true: note that knowing those sectional curvatures entails knowing the coefficients $R_{ijij}$ of the Riemann curvature tensor (and the other coefficients obtained through the usual relations). These, however, are not enough to determine the whole curvature tensor $R$ (remember that knowing the sectional curvature is equivalent to knowing the curvature tensor $R$).

Nonetheless, if we suppose $\dim M\ge3$ and $K$ constant for every $2$-plane at every point $m$, then $K$ is costant throughout the manifold (see Riemannian Geometry, Gallot,Hulin,Lafontaine, 3.10).