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For a compact manifold embedded in Euclidean space, the restricted linear height is Morse for generic directions

Statement

Let $M \subseteq R^{N}$ be a compact embedded smooth manifold.

If $N = 1$ , then every nonzero linear functional on $R$ restricts to a Morse function on $M$ .
If $N \geq 2$ , then there is a null subset $E \subseteq S^{N - 1}$ such that for every $u \in S^{N - 1} ∖ E$ , the height function $h_{u} : M \to R, h_{u} (x) = u \cdot x$ is Morse.

Thus restricted linear heights are Morse for generic directions.

Facts & Assumptions

Given: A compact embedded smooth manifold $M \subseteq R^{N}$ .

[F1]

A smooth function is Morse exactly when its differential section is transverse to the zero section (A smooth function is Morse if and only if its differential section is transverse to the zero section).

[L1]

Parametric transversality makes the bad parameter set null when the total family is transverse to the target submanifold (Parametric transversality).

[L2]

The zero section of the cotangent bundle is an embedded submanifold (The zero section is a smooth embedding).

[A1]

For $u \in S^{N - 1}$ and $x \in M$ , the differential of $h_{u}$ at $x$ is the cotangent vector $v \mapsto u \cdot v$ on $T_{x} M$ . If $d (h_{u})_{x} = 0$ , then $T_{x} M \subseteq u^{⊥} = T_{u} S^{N - 1}$ , so varying the parameter $u$ in tangent directions to the sphere produces every cotangent vector on $T_{x} M$ .

Proof

technique · direct

1.1

If $N = 1$ , then every embedded compact submanifold of $R$ is zero-dimensional, hence finite. The restriction of a nonzero linear functional to a finite manifold has all points critical and nondegenerate in the zero-dimensional Morse convention, so it is Morse.

givenalgebra

1.2

Assume now that $N \geq 2$ . Define a smooth family of cotangent sections by $D : M \times S^{N - 1} \to T^{*} M, D (x, u) = d (h_{u})_{x} .$ By [L2], its target zero section is an embedded submanifold. At any zero $(x, u)$ , [A1] says that the parameter derivative in tangent directions to $S^{N - 1}$ spans the whole fibre $T_{x}^{*} M$ , so $D$ is transverse to the zero section.

L2A1givenconstruct

2.1

Applying [L1] to the family $D$ shows that the set $E \subseteq S^{N - 1}$ of directions for which $d (h_{u})$ fails to be transverse to the zero section is null. For every $u \notin E$ , [F1] makes $h_{u}$ a Morse function on $M$ .

F1L1step 1.2

3.1

Step 1.1 handles $N = 1$ , and step 2.1 handles $N \geq 2$ . Therefore restricted linear heights are Morse for generic directions.

step 1.1step 2.1∎

Depends on

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Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 10 (standard reference, not scraped)
Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)