Morse–Bott function

Remarks

The Hessian is defined at critical points as in the definition of a nondegenerate critical point. Since $f$ is constant on each component $C$, the Hessian always vanishes on $T_p C$; the second condition says it is nondegenerate on the normal space $T_p M / T_p C$. The index of a critical component is the number of negative directions of the Hessian on this normal space. A Morse function is a Morse–Bott function whose critical components are points.

Example: $f(x, y) = x^2$ on $\R^2$ has $\operatorname{Crit}(f) = \set{x = 0}$, a line. At $(0, y)$ the Hessian matrix is $\operatorname{diag}(2, 0)$, whose kernel is spanned by $\partial_y$, the tangent direction of the line; so $f$ is Morse–Bott, with critical line of index $0$.

Non-example: $g(x, y) = x^4$ also has $\operatorname{Crit}(g) = \set{x = 0}$, a submanifold, but its Hessian at $(0, y)$ is $0$. Its kernel is all of $\R^2$, not the tangent line, so the second condition fails. Another non-example is $h(x, y) = x^2 y^2$, whose critical set is the union of the two axes, which is not a submanifold near the origin.