nLab regular value

Redirected from "critical values".
Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A regular value of a map ff between smooth manifolds is an element of the codomain of ff that is not a critical value, i.e., not the image of a critical point of ff.

Definition

Definition

(regular value)

For

f:XY f \;\colon\; X \longrightarrow Y

a differentiable function between differentiable manifolds (e.g. a smooth function between smooth manifolds) a point qf(X)Yq \in f(X) \subset Y in the image of ff is called a regular value of ff if at all points pf 1({q})p \in f^{-1}(\{q\}) in its preimage, the differential

df p:T pXT f(x)Y=T qY d f_p \;\colon\; T_p X \longrightarrow T_{f(x)} Y = T_{q}Y

is a surjective function between the corresponding tangent spaces.

A function all whose values are regular values is called a submersion.

(e.g. Kosinski 93, II (2.4))

Remark

(relation to transversality)

That qYq \in Y is a regular value (Def. ) of f:XYf \colon X \to Y means equivalently that ff is a transverse map to the submanifold-inclusion *qY\ast \overset{q}{\hookrightarrow} Y.

In this sense transversality generalizes the concept of regular values.

Properties

Inverse image

The inverse function theorem implies that:

Proposition

The inverse image f 1(q)Xf^{-1}(q) \subset X of a smooth function f:XYf \colon X \to Y at a regular value qYq \in Y is a smooth manifold of XX.

Together with Thom's transversality theorem, this is the key to the proof of the Pontryagin-Thom isomorphism.

References

Last revised on July 18, 2022 at 05:07:46. See the history of this page for a list of all contributions to it.