PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The diagonal is an embedded submanifold

Statement

For every smooth manifold $M$ , the diagonal

$Δ_{M} : = {(p, p) : p \in M} \subseteq M \times M$

is an embedded submanifold of dimension $\dim M$ .

Facts & Assumptions

Given: A smooth manifold $M$ .

[F1]

Embedded submanifolds are characterized by slice charts (Embedded submanifolds and slice charts).

[L1]

$M \times M$ has the canonical product smooth structure (Products of smooth manifolds have a canonical product smooth structure).

[L2]

Chart maps are diffeomorphisms onto open Euclidean sets (Chart maps are diffeomorphisms onto Euclidean open sets).

Proof

technique · direct

1.1L1L2given

Fix $(p, p) \in Δ_{M}$ . Choose a chart $φ : U \to Ω \subseteq R^{m}$ at $p$ . By [L1] and [L2], the product chart $φ \times φ : U \times U \to Ω \times Ω$ is a smooth chart on $M \times M$ .

2.1F1step 1.1algebra

In coordinates $(u, v) \in Ω \times Ω$ , the diagonal becomes ${(u, v) : u = v}$ . The linear change of variables $L (u, v) : = (u, v - u)$ is a diffeomorphism from $Ω \times Ω$ onto the open set $L (Ω \times Ω)$ , and $L ({(u, u) : u \in Ω}) = L (Ω \times Ω) \cap (R^{m} \times {0}) .$ Therefore $Δ_{M}$ is locally a coordinate slice and hence an embedded submanifold by [F1].

3.1step 2.1∎

The slice has dimension $m = \dim M$ , so the diagonal has that dimension as an embedded submanifold.

Depends on

Used by

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, Embedded Submanifolds (standard reference, not scraped)