The critical Hessian on a manifold #
The Hessian of a real function f at a critical point x of a manifold is a symmetric bilinear
form on the tangent space TangentSpace I x. Here it is defined as the second derivative of the
coordinate expression of f in the preferred extended chart at x; the tangent space at x is
the model vector space, so this is a continuous bilinear map on TangentSpace I x.
Away from critical points this second derivative depends on the chart (on a Riemannian manifold,
on the connection), because the second-order chain rule carries a first-order term. At a critical
point that term vanishes. The central result of this file is that, on a boundaryless manifold,
the critical Hessian of a C² function is then the pull-back of the Hessian computed in any
chart of the C² maximal atlas along the manifold derivative of that chart. In particular
nondegeneracy and the Morse index of a critical point may be computed in any such chart, and the
Morse index is the negative index of inertia of the critical Hessian.
Main declarations #
TauCeti.criticalHessian: the Hessian at a point, as a bilinear form on its tangent space.TauCeti.fderiv_fderiv_comp_chartAt_symm: on a manifold modelled on a real normed space, the second derivative in the preferred chart is the critical Hessian.TauCeti.criticalHessian_apply_eq_of_mem_maximalAtlas: at a critical point the critical Hessian is the Hessian in an arbitrary chart of the maximal atlas, composed with the chart derivative.TauCeti.criticalHessian_apply_swap: the critical Hessian of aC²function is symmetric.TauCeti.criticalHessian_apply_eq_mvfderiv: at a critical point, differentiatingdf(Y)computes the Hessian against the value of the vector fieldY.TauCeti.mvfderiv_mvfderiv_extend_eq_criticalHessian: at any point, differentiatingdf(W)alongv, forWthe vector field with constant coordinateswin the preferred chart, computes the HessiancriticalHessian I f x v w.TauCeti.criticalHessian_mfderiv_apply_eq_zero: the critical Hessian vanishes on directions tangent to a family of critical points.TauCeti.criticalHessian_comp_snd_apply: the critical Hessian of(x, v) ↦ q vonM × Vis the second derivative ofqon the second components.TauCeti.criticalHessian_mfderiv_eq_zeroandTauCeti.range_mfderiv_le_radical_criticalHessian: tangent vectors to a family of critical points lie in the kernel of the critical Hessian, which is the radical of its quadratic form (TauCeti.radical_toQuadraticMap_toBilinForm_criticalHessian).TauCeti.isManifoldNondegenerateCriticalPoint_iff_criticalHessian: an intrinsic characterization of nondegenerate critical points.TauCeti.isManifoldNondegenerateCriticalPoint_iff_of_mem_maximalAtlasandTauCeti.manifoldMorseIndex_eq_morseIndex_of_mem_maximalAtlas: nondegeneracy and the Morse index in an arbitrary chart.TauCeti.eventually_manifoldMorseIndex_le: the Morse index is lower semicontinuous on the critical set, read off fromTauCeti.eventually_morseIndex_lein the preferred chart.TauCeti.isMorseOn_iff_of_mem_maximalAtlas: being Morse on a subset of a chart source, read in that chart.TauCeti.IsManifoldNondegenerateCriticalPoint.manifoldMorseIndex_eq_zero_iffandTauCeti.manifoldMorseIndex_eq_finrank_iff: definiteness tests for the extreme indices.Module.Basis.criticalHessianMatrix: the Gram matrix of the critical Hessian in a tangent-space basis, with determinant, inertia, and definiteness comparisons (Module.Basis.criticalHessianMatrix_posDef_iffandModule.Basis.neg_criticalHessianMatrix_posDef_iff).TauCeti.exists_negDef_subspace_finrank_eq_manifoldMorseIndex: the Morse index is attained by a negative-definite tangent subspace.TauCeti.exists_posDef_subspace_finrank_eq_manifoldMorseIndex_neg: the Morse index of-fis attained by a positive-definite tangent subspace.
References #
- J. Milnor, Morse Theory, Princeton University Press, 1963, §2.
- M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 1.
The critical Hessian of a real function f at a point x of a manifold: the second
derivative at extChartAt I x x of the coordinate expression of f in the preferred extended
chart, as a bilinear form on the tangent space at x.
Like TauCeti.manifoldMorseIndex, the definition makes sense at every point and carries no
regularity hypotheses. At a critical point of a C² function on a manifold without boundary it is
intrinsic: TauCeti.criticalHessian_apply_eq_of_mem_maximalAtlas computes it in an arbitrary chart
there. At a noncritical point it depends on the preferred chart, and on a manifold with boundary
or corners it may also depend on the chart inverse outside range I.
Instances For
The critical Hessian is the second derivative of the preferred coordinate expression.
On a manifold modelled on a real normed space, the second derivative in the preferred chart is the critical Hessian.
The diagonal of the critical Hessian is the Hessian quadratic form of the preferred coordinate expression.
The quadratic form of the critical Hessian is the Hessian quadratic form of the preferred coordinate expression.
The Morse index is the negative index of inertia of the critical Hessian.
The critical Hessian depends only on the germ of the function.
Negating the function negates the critical Hessian.
The Morse index of -f is the positive inertia index of the critical Hessian of f.
On a normed space with its self-model, the critical Hessian is the second derivative.
At a nondegenerate critical point the critical Hessian is invertible as a map to the dual of the tangent space.
On a nonzero model space, the critical Hessian at a nondegenerate critical point is nonzero: a critical point with vanishing Hessian is degenerate.
On a finite-dimensional model space, an injective critical Hessian is invertible as a map to the dual of the tangent space.
Matrix and subspace descriptions #
The matrix of the critical Hessian of f in a basis of the tangent space.
Instances For
The entries of the critical Hessian matrix are its values on basis vectors.
The critical Hessian matrix depends only on the germ of the function.
Negating the function negates its critical Hessian matrix.
Critical Hessian matrices in two tangent-space bases are related by congruence.
The critical Hessian is recovered from its matrix by the usual coordinate formula.
The negative inertia index of the critical Hessian matrix is the manifold Morse index.
The critical Hessian is invertible exactly when its matrix in any basis has nonzero determinant.
Restricting the critical Hessian to a tangent subspace cannot increase its negative inertia index.
A tangent subspace on which the critical Hessian is negative definite has dimension at most the Morse index.
There is a negative-definite tangent subspace whose dimension is the Morse index.
A tangent subspace on which the critical Hessian is positive definite has dimension at most
the Morse index of -f.
There is a positive-definite tangent subspace whose dimension is the Morse index of -f.
A tangent subspace on which the critical Hessian is positive definite has dimension at most the dimension of the manifold minus the Morse index.
At a nondegenerate critical point there is a positive-definite tangent subspace of dimension
the dimension of the manifold minus the Morse index. For a space that is not finite, the dimension
is 0 by convention and the index is 0.
At a nondegenerate critical point the Morse index vanishes exactly when the critical Hessian is positive definite.
The Morse index is the dimension exactly when the critical Hessian is negative definite.
The critical Hessian of q ∘ Prod.snd at a point (x, y) is the second derivative of q at
y on the second components.
Boundaryless manifolds: symmetry and chart independence #
The critical Hessian of a function which is C² at x is symmetric.
The bilinear form of the critical Hessian of a function which is C² at x is symmetric.
The radical of the quadratic form of the critical Hessian of a C² function is the kernel of
the critical Hessian: by symmetry, a tangent vector is orthogonal to every vector exactly when it
is a null vector orthogonal to every vector.
A nondegenerate critical point on a boundaryless manifold is a point where the function is
C², its differential vanishes, and its critical Hessian is invertible.
The matrix of the critical Hessian of a C² function is symmetric.
The matrix of the critical Hessian of a C² function is positive definite exactly when the
critical Hessian is positive definite.
The negated matrix of the critical Hessian of a C² function is positive definite exactly when
the critical Hessian is negative definite.
In a tangent-space basis, nondegeneracy of a critical point is equivalent to C² regularity,
vanishing differential, and a nonzero determinant of the critical Hessian matrix.
At a critical point, differentiating df(Y) in a tangent direction computes the critical
Hessian against Y x. The derivative of the vector field contributes nothing because df = 0.
In particular this expression depends only on the value of Y at the point.
At every point, critical or not, differentiating df(W) along v, where
W = FiberBundle.extend E w is the vector field with constant coordinates w in the preferred
chart at x, gives criticalHessian I f x v w.
The critical Hessian vanishes along the critical set. If a map γ, differentiable at s,
sends every point near s to a critical point of a function f which is C² at γ s, then the
critical Hessian of f at γ s vanishes on the image of the derivative of γ.
The critical Hessian vanishes along the critical set, in the second argument.
The critical Hessian vanishes along a family of critical points. If a differentiable map
i sends every point near s to a critical point of a function f which is C² at i s, then
every tangent vector to i at s lies in the kernel of the critical Hessian at i s.
The tangent directions of a family of critical points lie in the radical of the critical Hessian.
The critical Hessian in an arbitrary chart. At a critical point of a C² function on a
boundaryless manifold, the critical Hessian is the Hessian of the coordinate expression in any
chart e of the C² maximal atlas, evaluated on the images of the tangent vectors under the
derivative of e. In particular it does not depend on the choice of chart.
Nondegeneracy in an arbitrary chart. On a boundaryless manifold, x is a nondegenerate
critical point of f exactly when the coordinate expression of f in any chart e of the C²
maximal atlas has a nondegenerate critical point at the image of x.
A nondegenerate critical point of the coordinate expression in the target of a chart of the
C² maximal atlas gives a nondegenerate critical point at its inverse image.
The Morse index in an arbitrary chart. At a critical point of a C² function on a
boundaryless manifold, the Morse index is the Morse index of the coordinate expression in any chart
e of the C² maximal atlas.
At a nondegenerate critical point in a chart's target, the coordinate Morse index equals the manifold Morse index at its inverse image.
Morse in a chart. On a boundaryless manifold, a function f smooth at every point of a
subset s of the source of a chart e of the C² maximal atlas is Morse on s exactly when its
coordinate expression f ∘ (e.extend I).symm has only nondegenerate critical points on
e.extend I '' s.
The critical Hessian in the tangent coordinates of a chart. Let x be a critical point
of a C² function f in the source of the preferred chart at x₀. On the tangent vectors at x
whose coordinates in the canonical tangent-bundle trivialization at x₀ are a and b, the
critical Hessian is the second derivative at x of the coordinate expression of f in the
preferred extended chart at x₀, applied to a and b.
The Morse index is lower semicontinuous on the critical set. On a boundaryless
finite-dimensional manifold, if f is C² at x, then at every critical point of f near x
the Morse index is at least the Morse index at x. The point x need not be critical, nor
nondegenerate.