Riemannian and intrinsic Hessians

For a metric-compatible connection ∇ and a function f which is C² at a point x, the covariant derivative of the gradient of f, paired with the metric, is the critical Hessian of f at x, the second derivative of f in the preferred chart at x, corrected by a first-order term:

⟪∇_v grad f, w⟫ = criticalHessian f x v w - df_x(∇_v W),

where W = FiberBundle.extend E w is the vector field with constant coordinates w in the preferred chart at x. The correction ∇_v W is the Christoffel map of ∇ in that chart (TauCeti.Manifold.covariantDerivative_extend_eq_christoffelMap), so away from critical points the connection, and for the Levi–Civita connection the metric, enters the Hessian through the term df_x(Γ(w, v)). Away from critical points both terms on the right depend on the preferred chart; only their difference is coordinate-free. No torsion assumption is needed.

At a critical point df_x = 0 and the correction vanishes: the pairing is the critical Hessian for every metric-compatible connection. In particular the Riemannian Hessian defined using the Levi–Civita connection is the critical Hessian there, so its value at a critical point is independent of the chosen metric and connection, and the intrinsic Morse index can be read from it.

The comparison requires only C² regularity of the function at the point and a C¹ metric.

Main results

• TauCeti.inner_covariantDerivative_mgradient_eq_criticalHessian_sub: the Hessian of f for a metric-compatible connection is the critical Hessian minus df_x(∇_v W).
• TauCeti.Manifold.hessian_eq_criticalHessian_sub: the same for the Riemannian Hessian.
• TauCeti.criticalHessian_apply_eq_inner_covariantDerivative and TauCeti.Manifold.hessian_eq_criticalHessian: at a critical point, these Hessians are the critical Hessian.

References

• J. Milnor, Morse Theory, Princeton University Press, 1963, §2.
• M. Audin and M. Damian, Morse Theory and Floer Homology, Springer, 2014, Chapter 1.
• J. M. Lee, Introduction to Riemannian Manifolds, GTM 176, 2nd ed., 2018, Chapter 4, for the Hessian and the Christoffel symbols of a connection in a chart.

theorem TauCeti.inner_covariantDerivative_mgradient_eq_criticalHessian_sub {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 2 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I 1 E fun (x : M) => TangentSpace I x] {f : M → ℝ} {x : M} (hf : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 f x) {cov : CovariantDerivative I E (TangentSpace I)} (hcov : cov.IsMetricCompatible) (v w : TangentSpace I x) : inner ℝ ((↑cov (Manifold.mgradient I f) x) v) w = ((criticalHessian I f x) v) w - (d% f x) ((↑cov (FiberBundle.extend E w) x) v)

The Hessian of a metric-compatible connection in the preferred chart. For any metric-compatible connection ∇ and any point x at which f is C², pairing ∇_v grad f with w gives the critical Hessian of f at x, minus df_x(∇_v W) for the vector field W with constant coordinates w in the preferred chart at x. That correction is df_x of the Christoffel map of ∇ in this chart, and vanishes at critical points. No torsion condition is needed.

theorem TauCeti.criticalHessian_apply_eq_inner_covariantDerivative {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 2 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I 1 E fun (x : M) => TangentSpace I x] {f : M → ℝ} {x : M} (hf : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 f x) (hcrit : d% f x = 0) {cov : CovariantDerivative I E (TangentSpace I)} (hcov : cov.IsMetricCompatible) (v w : TangentSpace I x) : ((criticalHessian I f x) v) w = inner ℝ ((↑cov (Manifold.mgradient I f) x) v) w

For any metric-compatible connection, the derivative of the gradient at a critical point, paired with the metric, equals the intrinsic critical Hessian. No torsion condition is needed.

theorem TauCeti.Manifold.hessian_eq_criticalHessian_sub {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 2 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I 1 E fun (x : M) => TangentSpace I x] {f : M → ℝ} {x : M} (hf : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 f x) (v w : TangentSpace I x) : hessian I f x v w = ((criticalHessian I f x) v) w - (d% f x) ((↑(CovariantDerivative.leviCivitaConnection I M) (FiberBundle.extend E w) x) v)

The Riemannian Hessian in the preferred chart. At any point x at which f is C², the Riemannian Hessian for a C¹ metric is the critical Hessian, the second derivative of f in the preferred chart at x, minus df_x(∇_v W) for the Levi–Civita connection ∇ and the vector field W with constant coordinates w in that chart.

theorem TauCeti.Manifold.hessian_eq_criticalHessian {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 2 M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I 1 E fun (x : M) => TangentSpace I x] {f : M → ℝ} {x : M} (hf : ContMDiffAt I (modelWithCornersSelf ℝ ℝ) 2 f x) (hcrit : d% f x = 0) (v w : TangentSpace I x) : hessian I f x v w = ((criticalHessian I f x) v) w

At a critical point, the Riemannian Hessian for any C¹ metric is the intrinsic Hessian. Only C² regularity of the function at this point is required.