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The intrinsic Hessian of a smooth function at a critical point
Definition
Let be smooth, and let be a critical point of (Critical points and critical values of a smooth function).
Choose a smooth chart around , write , and let . The matrix
is symmetric, and it defines a bilinear form on by
By Critical-point Hessian matrices transform by congruence under chart changes ↗, this bilinear form is independent of the chosen chart. It is the Hessian of at .
Depends on
Used by
- A Morse gradient zero contributes (-1)^λ to the index Corollary
- A degenerate critical set can be nonisolated Counterexample
- An isolated critical point can be degenerate Counterexample
- Euler equality alone does not imply perfectness Counterexample
- Nondegenerate critical points, nullity, index, and coindex Definition
- A created cancelling pair contributes a (1+t)tᵏ term Example
- A Morse function on the torus is perfect over every field Example
- Real projective space shows coefficient-dependent perfectness Example
- The height function on a sphere is perfect Example
- A smooth function is Morse if and only if its differential section is transverse to the zero section Lemma
- At a critical point, the intrinsic Hessian agrees with the Levi-Civita Hessian Lemma
- Critical-point Hessian matrices transform by congruence under chart changes Lemma
- Nondegenerate critical points are isolated Lemma
- Splitting one Morse coordinate preserves the residual Hessian Lemma
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)
- Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)