formal immersion of smooth manifolds


Étale morphisms

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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

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

Write Imm f(M,N)Imm^f(M,N) for the space of such formal immersions. There is a fibration over the space of smooth functions from MM to NN, Map sm(M,N)Map^{sm}(M, N), forgetting the bundle homomorphism, whose fiber over ff is Hom Vect M inj(TM,f *TN)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)Map^{sm}(M, N) is homotopy equivalent to the space of all continuous functions, Map(M,N)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 dfd f, there is a natural continuous function Imm(M,N)Imm f(M,N)Imm(M,N) \to Imm^f(M,N), sending an actual immersion ff to the formal immersion with injective bundle morphism dfd f.

Stephen Smale and Morris Hirsch established that when MM is compact, and also either MM is open (in the sense that the complement of the boundary has no compact component) or dim(M)<dim(N)dim(M) \lt dim(N), then the map Imm(M,N)Imm f(M,N)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, n+k)Map(S k,V k( n+k))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( n+k)V_k(\mathbb{R}^{n+k}) is the Stiefel manifold of kk-frames in n+k\mathbb{R}^{n+k}, the previous result establishes that isotopy classes of immersions of S kS^k into n+k\mathbb{R}^{n+k} are in bijection with π kV k( n+k)\pi_k V_k(\mathbb{R}^{n+k}). In the case where k=2k = 2 and n=1n = 1, we find that immersions of the 2-sphere into 3\mathbb{R}^3 are classified by π 2V 2( 3)=π 2(SO(3))=0\pi_2 V_2(\mathbb{R}^3)= \pi_2(SO(3))= 0, in other words, all such immersions are isotopic. In particular, S 2S^2 can be turned inside-out (sphere eversion) inside R 3R^3 by moving through a family of immersions.