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For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or ...
A less trivial example of a degenerate critical point is the origin of the monkey saddle. The index of a non-degenerate critical point of is the dimension of the largest subspace of the tangent space to at on which the Hessian is negative definite.
The Picard–Lefschetz formula describes the monodromy at a critical point. Suppose that f is a holomorphic map from an (k+1)-dimensional projective complex manifold to the projective line P 1. Also suppose that all critical points are non-degenerate and lie in different fibers, and have images x 1,...,x n in P 1. Pick any other point x in P 1.
Otherwise it is non-degenerate, and called a Morse critical point of . The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4]
The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each.
The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. When the degenerate points are not merely accidental, but are structurally stable , the degenerate points exist as organising centres for particular geometric structures of lower degeneracy ...
The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space.
A pseudo-Riemannian manifold (M, g) is a differentiable manifold M that is equipped with an everywhere non-degenerate, smooth, symmetric metric tensor g. Such a metric is called a pseudo-Riemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.