Transverse intersections of unstable and stable manifolds

Let p and q be nondegenerate critical points of a globally C² function f with globally Lipschitz gradient on a finite-dimensional real inner product space. The points of W^u(p) ∩ W^s(q), the intersection of the unstable set of p with the stable set of q, are the points lying on trajectories of the negative-gradient flow running from p to q. The Morse–Smale condition asks that these two embedded submanifolds meet transversally: at every common point y, their tangent spaces span the ambient space. Tangent spaces are taken intrinsically, as spans of the tangent cones tangentConeAt ℝ _ y.

This file proves the two basic consequences of transversality at a point y of W^u(p) ∩ W^s(q).

• Near y, the intersection is an embedded C¹ submanifold of dimension morseIndex f p - morseIndex f q: a C¹ chart with C¹ inverse flattens it onto the intersection of the two tangent spaces, which is its own tangent space at y, of dimension d with d + morseIndex f q = morseIndex f p. This is the set of points on connecting trajectories, each trajectory counted once for every one of its points (the parametrized trajectory locus). The space of unparametrized trajectories, whose points the Morse differential counts, is modelled by a level slice of this locus, of dimension one fewer: see TauCeti.Analysis.Calculus.Morse.TrajectorySpace.
• If p ≠ q, the Morse index drops strictly: morseIndex f q < morseIndex f p. The velocity -∇f y of the trajectory through y is tangent to both invariant sets and is nonzero, so the intersection has positive dimension.

The tangent spaces themselves have the expected dimensions: morseIndex f p for the unstable set of p, and the complementary dimension for the stable set of q.

Main results

• TauCeti.neg_gradient_mem_tangentConeAt_unstableSet_inter_stableSet: the velocity of a connecting trajectory is tangent to W^u(p) ∩ W^s(q).
• TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_unstableSet and TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_stableSet_add_morseIndex: the dimensions of the tangent spaces of the unstable and stable sets.
• TauCeti.IsNondegenerateCriticalPoint.exists_unstableSet_inter_stableSet_chart: a transverse intersection of an unstable and a stable set is an embedded C¹ submanifold, flattened onto the intersection of the tangent spaces.
• TauCeti.IsNondegenerateCriticalPoint.span_tangentConeAt_unstableSet_inter_stableSet: the tangent space of the intersection is the intersection of the tangent spaces.
• finrank_span_tangentConeAt_unstableSet_inter_stableSet_add_morseIndex (in the namespace TauCeti.IsNondegenerateCriticalPoint): the intersection has dimension morseIndex f p - morseIndex f q.
• TauCeti.IsNondegenerateCriticalPoint.morseIndex_lt_of_mem_unstableSet_inter_stableSet: along a transverse trajectory joining distinct critical points, the Morse index drops strictly.

References

• M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 2 (Morse–Smale condition) and Chapter 3 (spaces of trajectories).

theorem TauCeti.neg_gradient_mem_tangentConeAt_unstableSet_inter_stableSet {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hf : LipschitzWith K (gradient f)) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) : -gradient f y ∈ tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q) y

The velocity -∇f y of the negative-gradient trajectory through a point y of W^u(p) ∩ W^s(q) is tangent to W^u(p) ∩ W^s(q), since the trajectory stays in this invariant set.

theorem TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_unstableSet {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p y : E} {K : NNReal} (hp : IsNondegenerateCriticalPoint f p) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hy : y ∈ (negativeGradientFlow f hf).unstableSet p) : Module.finrank ℝ ↥(Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y)) = morseIndex f p

The tangent space of the unstable set of a Morse critical point p, at any of its points, has dimension the Morse index of p.

theorem TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_stableSet_add_morseIndex {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {q y : E} {K : NNReal} (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hy : y ∈ (negativeGradientFlow f hf).stableSet q) : Module.finrank ℝ ↥(Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y)) + morseIndex f q = Module.finrank ℝ E

The tangent space of the stable set of a Morse critical point q, at any of its points, has dimension the ambient dimension minus the Morse index of q.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_unstableSet_inter_stableSet_chart {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) : ∃ (e : OpenPartialHomeomorph E E), y ∈ e.source ∧ (∀ z ∈ e.source, ContDiffAt ℝ 1 (↑e) z) ∧ (∀ z ∈ e.target, ContDiffAt ℝ 1 (↑e.symm) z) ∧ ∀ z ∈ e.source, z ∈ (negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q ↔ ↑e z ∈ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊓ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y)

Transverse unstable and stable sets meet in an embedded submanifold. If the unstable set of a Morse critical point p and the stable set of a Morse critical point q meet transversally at y, then near y their intersection, the set of points lying on trajectories from p to q, is flattened by a C¹ chart with C¹ inverse onto the intersection of their tangent spaces.

theorem TauCeti.IsNondegenerateCriticalPoint.span_tangentConeAt_unstableSet_inter_stableSet {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q) y) = Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊓ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y)

The tangent space of a transverse intersection. If the unstable set of a Morse critical point p and the stable set of a Morse critical point q meet transversally at y, then the tangent space of W^u(p) ∩ W^s(q) at y is the intersection of their tangent spaces.

theorem TauCeti.IsNondegenerateCriticalPoint.finrank_span_tangentConeAt_unstableSet_inter_stableSet_add_morseIndex {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) : Module.finrank ℝ ↥(Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p ∩ (negativeGradientFlow f hf).stableSet q) y)) + morseIndex f q = morseIndex f p

The intersection has dimension the index difference. If the unstable set of a Morse critical point p and the stable set of a Morse critical point q meet transversally at y, then the tangent space of W^u(p) ∩ W^s(q) at y has dimension morseIndex f p - morseIndex f q.

theorem TauCeti.IsNondegenerateCriticalPoint.morseIndex_lt_of_mem_unstableSet_inter_stableSet {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {p q y : E} {K : NNReal} (hp : IsNondegenerateCriticalPoint f p) (hq : IsNondegenerateCriticalPoint f q) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) (hpq : p ≠ q) (hyu : y ∈ (negativeGradientFlow f hf).unstableSet p) (hys : y ∈ (negativeGradientFlow f hf).stableSet q) (htr : Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).unstableSet p) y) ⊔ Submodule.span ℝ (tangentConeAt ℝ ((negativeGradientFlow f hf).stableSet q) y) = ⊤) : morseIndex f q < morseIndex f p

The Morse index drops strictly along a transverse trajectory. If a trajectory of the negative-gradient flow runs from a Morse critical point p to a different Morse critical point q, and the unstable set of p meets the stable set of q transversally at a point y of it, then morseIndex f q < morseIndex f p. Under the Morse–Smale condition, trajectories therefore only run from higher to lower index.