nLab
regular value

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

tangent cohesion

differential cohesion

graded differential 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}{}& ʃ &\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

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

References

Created on February 5, 2019 at 11:16:46. See the history of this page for a list of all contributions to it.