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Sylvester inertia makes the Morse index intrinsic
Statement
Let be smooth and let be a critical point of . Then the numbers of positive, negative, and zero directions of any chart matrix of are independent of the chart. Equivalently, the nullity, index, and coindex of are intrinsic.
Facts & Assumptions
Given: A smooth function and a critical point .
Hessian matrices in two charts are congruent (Critical-point Hessian matrices transform by congruence under chart changes).
Nullity, index, and coindex are defined from the Hessian as kernel dimension and maximal negative- and positive-definite dimensions (Nondegenerate critical points, nullity, index, and coindex).
Congruent real symmetric matrices have the same inertia data (Sylvester's law of inertia: every real symmetric form is congruent to , and is unique).
The inertia counts are exactly the numbers of positive, negative, and zero entries in a diagonal normal form (Positive and negative definiteness, the inertia , rank , and signature of a real symmetric bilinear or quadratic form).
Proof
By [F1], any two chart matrices of are congruent real symmetric matrices.
Therefore [L1] and [L2] give the same triple for every chart matrix.
For one diagonal normal form, the zero count is the kernel dimension, the negative count is the maximal dimension of a negative-definite subspace, and the positive count is the maximal dimension of a positive-definite subspace, by [F2] and [L2]. Hence nullity, index, and coindex are the same in every chart.
Thus the Morse nullity, index, and coindex are intrinsic.
Depends on
- Critical-point Hessian matrices transform by congruence under chart changes
- Nondegenerate critical points, nullity, index, and coindex
- Sylvester's law of inertia: every real symmetric form is congruent to $\operatorname{diag}(I_p,-I_q,0_r)$, and $(p,q,r)$ is unique
- Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form
Used by
- Morse lemma Theorem
Dependency tree · two levels
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Sources
- Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)
- Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)