I am trying to find out a non-degenerate, positive, bilinear form defined for every point $p$ in $S^2$, such that it is not a metric and illustrate the same (i.e. it must not be satisfying the smoothly varying criterion). Need an explicit example.
Then a further question would be :
i) How do I check if such a randomly given '$g$' will not be a Riemannian metric on a sphere ?
One direction of thought is : Let 'g' be a metric on a smooth manifold 'M' (here $S^2$), by the existence of a metric on a smooth manifold. Now consider for our case a chart $(U,\phi)$, and try with :
$$ \tilde{g} = (1 + \chi_U)g $$ This should be an example as on the boundary of $U$ it will not be smooth, but how do I explicitly show it in computation taking some particular vector fields ?