Math 2.0

Manifolds and cell complexes Definition

Nondegenerate critical point

Builds on Smooth manifoldSmooth functionCritical pointHessian matrixSymmetric bilinear formNondegenerate bilinear form

Definition

Let MM be a smooth manifold of dimension nn and let f ⁣:M→ℝf \colon M \to \R be a smooth function. Let pp be a critical point of ff, so dfp=0df_p = 0. The Hessian of ff at pp is the symmetric bilinear form on TpMT_p M given by Hess⁡pf(v,w)=v(Wf),\operatorname{Hess}_p f(v, w) = v\big(W f\big), where WW is any vector field with W(p)=wW(p) = w. The critical point pp is nondegenerate if Hess⁡pf\operatorname{Hess}_p f is a nondegenerate bilinear form. Otherwise pp is degenerate.

Remarks

Because dfp=0df_p = 0, the value v(Wf)v(Wf) does not depend on the extension WW of ww, and the form is symmetric. In a chart (x1,…,xn)(x_1, \dots, x_n) around pp, Hess⁡pf\operatorname{Hess}_p f has the matrix (∂2f/∂xi∂xj(p))\big(\partial^2 f / \partial x_i \partial x_j (p)\big), the Hessian matrix, and pp is nondegenerate if and only if this matrix is invertible.

Example: f(x,y)=x2+xy+y2f(x, y) = x^2 + xy + y^2 on ℝ2\R^2 has the critical point (0,0)(0, 0) with Hessian matrix (2112)\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}. Its determinant is 3≠03 \ne 0, so the origin is nondegenerate.

Non-example: g(x,y)=x2+y4g(x, y) = x^2 + y^4 has the critical point (0,0)(0, 0) with Hessian matrix (2000)\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}, which is singular, so the origin is degenerate.

Nondegenerate critical points are isolated among critical points. A function all of whose critical points are nondegenerate is a Morse function.

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Critical point, concept, no evidence levelCritical pointconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptNondegenerate bilinear form, concept, no evidence levelNondegenerate bilinea…conceptSmooth function, concept, no evidence levelSmooth functionconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSymmetric bilinear form, concept, no evidence levelSymmetric bilinear fo…conceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptHigher partial derivatives, concept, no evidence levelHigher partial deriva…conceptHessian matrix, concept, no evidence levelHessian matrixconceptNondegenerate critical point, definition, no evidence levelNondegenerate critica…definition
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  1. Critical point concept
  2. Euclidean space object
  3. Function concept
  4. Nondegenerate bilinear form concept
  5. Smooth function concept
  6. Smooth manifold concept
  7. Symmetric bilinear form concept
  8. Function of several variables concept
  9. Partial derivative concept
  10. Higher partial derivatives concept
  11. Hessian matrix concept
  12. Nondegenerate critical point definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Morse_theory
About this page
TagG43BR3YVKQ
URIm2://G43BR3YVKQ
Kinddefinition
WritingW0 stub
MSC57R70, 58E05, 58K05
DefinesNondegenerate critical point
UsesSmooth manifold, Smooth function, Critical point, Hessian matrix, Symmetric bilinear form, Nondegenerate bilinear form

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