Regular and critical points and values

Depends on

• Immersions, submersions, and constant-rank maps

Used by

• Zero-level symplectic reduction and the dimension formula Corollary
• Critical points and critical values of a smooth function Definition
• The critical locus and critical value set Definition
• The mod-two degree of a map to a sphere Definition
• A value outside the image can still be regular Example
• Maps from even projective space to the sphere use mod-two degree Example
• A regular value need not belong to the image False statement
• An infinitesimal moment map is automatically equivariant False statement
• Every integer is realized by a map to the sphere Lemma
• Smooth strict plurisubharmonic regularization of a psh exhaustion Lemma
• The mod-two degree is well defined and homotopy invariant Lemma
• The signed preimage count equals the degree Lemma
• Hessian of distance in terms of radial jacobi fields Proposition
• Regularity of a moment map is equivalent to local freeness Proposition
• The tangent space of a regular level set is the kernel Proposition
• Transversality to a point is the regular-value condition Proposition
• A regular level set is an embedded submanifold Theorem
• Marsden--Weinstein--Meyer symplectic reduction Theorem
• Morse-Sard for smooth manifolds Theorem
• Reduction in stages for free proper regular actions Theorem
• Regular-value formula for compact-support degree Theorem
• The Hopf mod-two degree theorem for nonorientable domains Theorem

Dependency tree · two levels

2 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, Ch. 5 (standard reference, not scraped)
• Will J. Merry, Differential Geometry, Definition 6.9 (standard reference, not scraped)
• Nigel Hitchin, Differentiable Manifolds, Definition 12 (standard reference, not scraped)