nLab formal immersion of smooth manifolds

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

A formal immersion $F$ of one smooth manifold, $M$, into another, $N$, is an injective bundle morphism $T M \to T N$ between their tangent bundles.

That is, $F$ consists of a smooth function $f \;\colon\; M \to N$ and a homomorphism of vector bundles $F \;\colon\; T M\to T N$ covering $f$ such that the linear function $F|_{x} \;\colon\; T_x M \to T_{f(x)} N$ is an injective function for every point $x$ in $M$.

Write $Imm^f(M,N)$ for the space of such formal immersions. There is a fibration over the space of smooth functions from $M$ to $N$, $Map^{sm}(M, N)$, forgetting the bundle homomorphism, whose fiber over $f$ is $Hom^{inj}_{Vect_M}(T M, f^{\ast}T N)$.

Note: Some authors define formal immersions in terms of continuous functions (e.g., Laudenbach17, p. 6). However, the space $Map^{sm}(M, N)$ is homotopy equivalent to the space of all continuous functions, $Map(M, N)$, due to integrating against a smoothing kernel.

Relation to immersions

Since an actual immersion of smooth manifolds is a formal immersion where the bundle morphism in question is specifically taken to be the pointwise derivative $d f$, there is a natural continuous function $Imm(M,N) \to Imm^f(M,N)$, sending an actual immersion $f$ to the formal immersion with injective bundle morphism $d f$.

Stephen Smale and Morris Hirsch established that when $M$ is compact, and also either $M$ is open (in the sense that the complement of the boundary has no compact component) or $dim(M) \lt dim(N)$, then the map $Imm(M,N) \to Imm^f(M,N)$ is a weak homotopy equivalence. This is an instance of the h-principle.

When combined with the result that $Imm^f(S^k,\mathbb{R}^{n+k}) \to Map(S^k, V_k(\mathbb{R}^{n+k}))$ can be shown to be a homotopy equivalence, where $V_k(\mathbb{R}^{n+k})$ is the Stiefel manifold of $k$-frames in $\mathbb{R}^{n+k}$, the previous result establishes that isotopy classes of immersions of $S^k$ into $\mathbb{R}^{n+k}$ are in bijection with $\pi_k V_k(\mathbb{R}^{n+k})$. In the case where $k = 2$ and $n = 1$, we find that immersions of the 2-sphere into $\mathbb{R}^3$ are classified by $\pi_2 V_2(\mathbb{R}^3)= \pi_2(SO(3))= 0$, in other words, all such immersions are isotopic. In particular, $S^2$ can be turned inside-out (sphere eversion) inside $R^3$ by moving through a family of immersions.