For concavity and convexity is that the Hessian be negative semidefinite and positive semidefinite But for strict concavity definiteness is only the sufficient condition not the necessary . What is the necessary condition for a two variable function to ensure strict concavity / convexity ?
Your initial statement is wrong: the Hessian does not have to exist for the function to be concave or convex. For example, $|x|$ is convex but does not have derivatives at $0$.
A necessary and sufficient condition for a convex/concave function to be strictly convex/concave is that its graph does not contain any line segment.