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Manifolds and cell complexes Definition

Morse index

Builds on Smooth manifoldSmooth functionCritical pointNondegenerate critical pointSymmetric bilinear form

Definition

Let MM be a smooth manifold, let f ⁣:M→ℝf \colon M \to \R be a smooth function, and let pp be a critical point of ff. Let H=Hess⁡pfH = \operatorname{Hess}_p f be its Hessian, the symmetric bilinear form on TpMT_p M from the definition of a nondegenerate critical point. The Morse index of ff at pp is the largest dimension of a linear subspace V⊆TpMV \subseteq T_p M such that H(v,v)<0H(v, v) < 0 for every nonzero v∈Vv \in V. The nullity of ff at pp is the dimension of the kernel {v∈TpM:H(v,w)=0 for all w∈TpM}\set{v \in T_p M : H(v, w) = 0 \text{ for all } w \in T_p M}.

Remarks

In a chart, the index is the number of negative eigenvalues of the Hessian matrix, counted with multiplicity, and the nullity is the dimension of its kernel. By Sylvester's law of inertia both are independent of the chart. The point pp is nondegenerate exactly when the nullity is 00, and the term Morse index is mostly used in that case.

Example: f(x,y,z)=x2−y2−3z2f(x, y, z) = x^2 - y^2 - 3z^2 on ℝ3\R^3 has Hessian matrix diag⁡(2,−2,−6)\operatorname{diag}(2, -2, -6) at the origin. It is negative definite on the yzyz-plane and not on any 33-dimensional subspace, so the index is 22 and the nullity is 00.

Non-example: for g(x)=x3g(x) = x^3 on ℝ\R, the critical point 00 has index 00 and nullity 11. Index 00 does not make 00 a local minimum here, since the point is degenerate; this is why the index describes the local shape only at nondegenerate critical points.

The same definition applies on a Hilbert manifold, where the index may be infinite. For the energy functional on paths, the index at a geodesic is computed by the Morse index theorem for geodesics.

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  1. Critical point concept
  2. Nondegenerate critical point concept
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  5. Symmetric bilinear form concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Morse_theory
About this page
TagAH059Y44NF
URIm2://AH059Y44NF
Kinddefinition
WritingW0 stub
MSC57R70, 58E05, 58E10
DefinesMorse index
UsesSmooth manifold, Smooth function, Critical point, Nondegenerate critical point, Symmetric bilinear form

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