On the Affine Gauss Maps of Submanifolds of Euclidean Space
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Abstract
It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface \(\mathcal{S}\) the set of oriented lines that cross \(\mathcal{S}\) orthogonally is a Lagrangian submanifold. Conversely, if \(\overline{\mathcal{S}}\) an ndimensional family of oriented lines is Lagrangian, there exists, locally, a 1parameter family of immersed, oriented, parallel hypersurfaces \(\mathcal{S}_t\) whose tangent spaces cross orthogonally the lines of \(\overline{\mathcal{S}}.\) The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold \(\mathcal{S}\) of \({\mathbb {R}^{}} ^m\) of dimension n and codimension \(k=mn,\) we may associate the affine kspace normal to \(\mathcal{S}\) at x. Conversely, given an ndimensional family \(\overline{\mathcal{S}}\) of affine kspaces of \({\mathbb {R}^{}} ^m\), we provide certain conditions granting the local existence of a family of ndimensional submanifolds \(\mathcal{S}\) which cross orthogonally the affine kspaces of \(\overline{\mathcal{S}}\). We also define a curvature tensor for a general family of affine spaces of \({\mathbb {R}^{}} ^m\) which generalizes the curvature of a submanifold, and, in the case of a 2dimensional family of 2planes in \({\mathbb {R}^{}} ^4\), show that it satisfies a generalized Gauss–Bonnet formula.
Keywords
Submanifolds in Euclidean nspace Gauss mapsMathematics Subject Classification
53A07 53B251 Introduction
It is well known that the space \(L({\mathbb {R}^{}} ^{n+1})\) of oriented lines of Euclidean space \({\mathbb {R}^{}} ^{n+1}\) enjoys a natural symplectic structure. The simplest way to understand this is through the identification of \(L({\mathbb {R}^{}} ^{n+1})\) with the tangent bundle of the unit sphere \(T {\mathbb {S}^{}} ^{n}.\) The canonical symplectic form of \(T{\mathbb {S}^{}} ^{n}\) is \(\omega = d \alpha , \) where \(\alpha \) is the tautological form (sometimes also called Liouville form) defined by the formula \(\alpha _{(p,v)} =\langle v, d\pi (.)\rangle \) at \((p,v)\in T{\mathbb {S}^{}} ^n\) (\(p\in {\mathbb {S}^{}} ^n,\)\(v\in T_p{\mathbb {S}^{}} ^n\)), where \(\pi : T{\mathbb {S}^{}} ^n \rightarrow {\mathbb {S}^{}} ^n\) is the canonical projection.
Moreover, given an immersed, oriented hypersurface \(\mathcal{S}\subset {\mathbb {R}^{}} ^{n+1},\) the set of oriented lines that cross \(\mathcal{S}\) orthogonally is a Lagrangian submanifold of \(L({\mathbb {R}^{}} ^{n+1})\). This fact has a nice geometric interpretation: generically, a nonflat hypersurface may be locally parametrized by its Gauss map; in this case the Lagrangian submanifold is a section of \(T{\mathbb {S}^{}} ^n\) and its generating function is the support function of the hypersurface. Conversely, given a Lagrangian submanifold \({\overline{\mathcal{S}}} \subset L({\mathbb {R}^{}} ^{n+1})\), there exists, locally, a 1parameter family of immersed, oriented, parallel hypersurfaces \(\mathcal{S}_t\) whose tangent spaces cross orthogonally the lines of \(\overline{\mathcal{S}}\). The situation is very similar if we replace the Euclidean space by a pseudoRiemannian space form \({\mathbb {Q}}^{n+1}_p\) of arbitrary signature \((p,n+1p)\) and \(L({\mathbb {R}^{}} ^{n+1})\) by the space of geodesics \(L({\mathbb {Q}}^{n+1}_p)\) of \({\mathbb {Q}}^{n+1}_p\) (see Anciaux 2014).
The aim of this paper is to generalize these facts to the higher codimension submanifolds of Euclidean (or pseudoEuclidean) space: to any point x of a submanifold \(\mathcal{S}\) of \({\mathbb {R}^{}} ^m\) of dimension n and codimension \(k=mn,\) we may associate the affine kspace normal to \(\mathcal{S}\) at x. We call this data the affine Gauss map of \(\mathcal{S}.\) Conversely, given an ndimensional family \(\overline{\mathcal{S}}\) of affine kspaces of \({\mathbb {R}^{}} ^m\) (a data that we call abstract affine Gauss map), it is natural to ask when it is the affine Gauss map of some submanifold; in other words: under which condition does there exist locally a family of ndimensional submanifolds \(\mathcal{S}\) which cross orthogonally the affine kspaces of \(\overline{\mathcal{S}}\)?
For this purpose, we first examine the Grassmannian \({\mathcal {Q}}\) of affine kspaces. It has a natural bundle structure and we define a natural 1form \(\alpha \) on it. This 1form is vectorvalued rather than realvalued whenever \(k>1\), and generalizes the classical tautological form of \(T{\mathbb {S}^{}} ^n.\)
Next, we consider a map \(\overline{\varphi } :\mathcal{M}\rightarrow {\mathcal {Q}}\) defined on an ndimensional manifold \(\mathcal{M}\). Pulling back the geometry of \({\mathcal {Q}}\) via the map \(\overline{\varphi }\) induces a natural bundle \(E_N\) of rank k over \(\mathcal{M}\), that we call abstract normal bundle, equipped with a natural connection \(\nabla ^{N}\) (see the next section for the precise definition). Our main result is the determination of some natural geometric conditions on \((E_N,\nabla ^{N})\) and \(\overline{\varphi }\) which are sufficient to ensure the existence of such ”integral” submanifolds \(\mathcal{S}.\) The known cases are easily recovered from our result: in the case of codimension \(k=1\), \({\mathcal {Q}}\) identifies to \({\mathbb {S}^{}} ^{n}\) and the integrability condition is equivalent to the vanishing of the symplectic form \(\omega \). If \(k >1\) and \((E_N,\nabla ^{N})\) is flat, the situation is quite similar.
We are also able to deal with the case of hypersurfaces in pseudoRiemannian spaceforms (Anciaux 2014), that we regard as submanifolds of pseudoEuclidean space \({\mathbb {R}^{}} ^{n+2}\) of codimension two contained in a (pseudo)sphere: in this case the map \(\overline{\varphi }\) is valued in the zero section of \({\mathcal {Q}}\), which makes the integrability condition easier to interpret.
A related problem consists of trying to reconstruct an immersed surface from its linear Gauss map. The difference is that instead of considering affine spaces, one deals with linear spaces, so less information is involved. This issue has been addressed for surfaces (\(n=2\)) in the following papers: Hoffman and Osserman (1983, 1985) and Weiner (1984, 1986b). To our knowledge this problem is open for higher dimensions \(n\ge 3.\)
Besides the problem of the prescription of the affine normal spaces of a submanifold, we study some geometric properties of a general congruence of affine spaces. Specifically, we propose a definition of its curvature, and, in the case of a congruence of planes in \({\mathbb {R}^{}} ^4\), we obtain a Gauss–Bonnet type formula. This generalizes to higher dimension and codimension similar results obtained in Guilfoyle and Klingenberg (2004) for a congruence of lines in \({\mathbb {R}^{}} ^3.\) Our treatment is simplified by the use of a formula expressing the curvature of the tautological bundles in terms of the Clifford product (Sect. 7; Appendix A); this formula might also be of independent interest.
The paper is organized as follows: in Sect. 2, we set some notation and state our main results. In Sect. 3, we introduce the generalized canonical tautological form, while Sects. 4 and 5 are devoted to the proofs of the main theorems. Section 6 deals with some special cases of low dimension and codimension. The last section is concerned with the general notion of curvature of a congruence. It establishes a Gauss–Bonnet type formula for a congruence of planes in \({\mathbb {R}^{}} ^4\). Two short appendices end the paper.
2 Notation and statement of results
Definition 1
Remark 1
If the abstract Gauss map \(\overline{\varphi }_o\) is in fact the Gauss map of an immersion of \(\mathcal{M}\) into \({\mathbb {R}^{}} ^m\) (assuming that an orientation on \(\mathcal{M}\) is given), the bundles \(E_T\) and \(E_N\) with their induced connections naturally identify to the tangent and the normal bundles of the immersion, with the Levi–Civita and the normal connections; see also Remark 12 in Sect. 7.
Theorem 1
Remark 2
We shall call s the support function of \(\varphi \). In the case of codimension one, i.e. \(E_N\) has rank one, s identifies with the usual support function of the immersed hypersurface \(\mathcal{S}\).
Remark 3
Theorem 2
Remark 4
Condition (7) is obviously necessary for the existence of an immersion \(\varphi \) with affine Gauss map \(\overline{\varphi },\) since, for \(\Phi =d\varphi ,\) it expresses the symmetry of the second fundamental form; see also Weiner (1984, 1986b) where this condition appears explicitly in the problem of finding immersions of surfaces with prescribed linear Gauss map.
If \({\mathcal{R}^{N}}=0,\) Condition (8) is equivalent to saying that \(\overline{\varphi }_o\) is an immersion, system (9) reduces to \(\overline{\varphi }^*\omega =0\) and we obtain the following result:
Corollary 1
Remark 5
This result generalizes the case of a congruence of oriented lines of \({\mathbb {R}^{}} ^{n+1}\) (Anciaux 2014): in this case \(E_N\) has rank 1, \(\nabla ^N\) is flat, so the assumptions of Corollary 1 are satisfied. Condition (10) then amounts to saying that \(\overline{\varphi }:\mathcal{M}\rightarrow {\mathcal {Q}}\simeq T{\mathbb {S}^{}} ^n\) is a Lagrangian map; see Sect. 3.
Remark 6
Remark 7
3 The Generalized Tautological Form
In this section we introduce a natural \(\tau _N\)valued oneform on the Grassmannian \({\mathcal {Q}}\) of the affine kspaces in \({\mathbb {R}^{}} ^m.\) We then show that it generalizes the classical tautological form on \(T{\mathbb {S}^{}} ^n,\) and finally that it also satisfies a tautological property.
3.1 The General Construction
3.2 Relation to the Classical Tautological Form
Here we explain how the form \(\alpha \) is actually a generalization of the canonical tautological form of \(T{\mathbb {S}^{}} ^{n}.\) We first observe that in the case \((n,k)=(m1,1),\) the hypersphere \({\mathbb {S}^{}} ^{n} \subset {\mathbb {R}^{}} ^{n+1}\) is identified with \({\mathcal {Q}}_o,\) the Grassmannian of oriented, linear hyperplanes of \({\mathbb {R}^{}} ^{n+1}\). It follows that \(T {\mathbb {S}^{}} ^{n}\) is identified with \({\mathcal {Q}},\) the Grassmannian of oriented, affine lines of \({\mathbb {R}^{}} ^{n+1}.\) Through these identifications, the natural projection \(\pi :T{\mathbb {S}^{}} ^{n}\rightarrow {\mathbb {S}^{}} ^{n}\) corresponds to the map \(\pi :{\mathcal {Q}}\rightarrow {\mathcal {Q}}_o\) defined in Sect. 3.1.
3.3 The Tautological Property
4 Proof of Theorem 1
5 Proof of Theorem 2
5.1 The Formal Resolution
A solution, if it exists, is not unique, but it is so modulo adding a parallel section \(\sum _{i=1}^r c_i s_i\) depending on r constants \(c_1,\ldots ,c_r\).
5.2 Existence of a Solution Which is an Immersion
Remark 8
For sake of simplicity, we first prove the result in the case of a flat normal bundle, which is the context of Corollary 1, and only give at the end of the section brief indications for the proof of the general case.
Lemma 1
Proof
Theorem 2 follows now from the next lemma:
Lemma 2
There exists \(\nu \in E_N\) such that \(B(\nu ):T\mathcal{M}\rightarrow E_T\) is an isomorphism.
Proof
The arguments above still hold if we replace \(E_N\) by \(Ker\ {\mathcal{R}^{N}}\subset E_N,\) noting that \(\overline{\varphi }_o^*\omega _o\in \Omega ^2(\mathcal{M},End(E_N))\) vanishes on \(Ker\ {\mathcal{R}^{N}}\) (thus extending Lemma 1), and that the assumption \(B(\nu )(X)=0, \forall \nu \in Ker\ {\mathcal{R}^{N}}\) is not possible if \(Im(d\overline{\varphi }_o(X))\) is not contained in \((Ker\ {\mathcal{R}^{N}})^\perp \) (for the proof of Lemma 2).
5.3 The Local Riemannian Foliation
Now, in the general case \(r <k\), the argument above still applies and we get the following result: if there exists a solution \(\varphi :\mathcal{M}\rightarrow {\mathbb {R}^{}} ^m\) which is an immersion at some point \(x_o\in \mathcal{M},\) then the set of solutions is a local Riemannian foliation of a submanifold of \({\mathbb {R}^{}} ^m\) of dimension \(n+r\) (the submanifold of \({\mathbb {R}^{}} ^m\) is the image of the local immersion \(\Phi ((c_1,\ldots ,c_r),x)=\varphi (x)+\sum _{i=1}^r c_is_i(x)\)).
6 Some Special Cases
6.1 Curves in Euclidean Space
In this section, we give more detail on the case of curves \(n=1\). According to Theorem 2, there is no integrability condition in this case. Hence in this case any abstract affine Gauss map \(\overline{\varphi }\) should actually be the Gauss map of a family of curves \(\varphi \).
Observe also that the Grassmannian of affine hyperplanes \({\mathcal {Q}} \) identifies with \({\mathbb {S}^{}} ^{m1} \times {\mathbb {R}^{}} \): at the pair \((\alpha , \lambda )\) we associate the affine hyperplane \(\lambda \alpha + \alpha ^{\perp }.\) Hence, given a curve \(\overline{\gamma }= (\alpha , \lambda ): I \rightarrow {\mathbb {S}^{}} ^{m1} \times {\mathbb {R}^{}} \), we claim that there exists a \((m1)\)parameter family of curves \(\gamma : I \rightarrow {\mathbb {R}^{}} ^m\) such that \(\forall t \in I, \gamma (t) \in \overline{\gamma }\) and \(\gamma ' \) is orthogonal to \(\overline{\gamma }\), i.e. \(\gamma '\) is collinear to \(\alpha .\)
Since \(\gamma '=(\lambda 'A) \alpha \), the curve \(\gamma \) fails to be regular precisely if \(A=\lambda '\). Clearly, it may happen only for a particular choice of the initial condition in the system \((*)\). Hence, except for a discrete set of values, the curves of the 2parameter family of solutions are immersed.
Remark 9
Note that we are in fact here in a case where Corollary 1 holds: all the hypotheses of the corollary are trivially satisfied since \(n=1.\)
6.2 Hypersurfaces in Space Forms
Lemma 3
Let \(\varphi : \mathcal{M}\rightarrow {\mathbb {R}^{}} ^{m}\) an immersion and assume \(\mathcal{M}\) is connected. Denote \(\overline{\varphi }: \mathcal{M}\rightarrow {\mathcal {Q}}\) its affine Gauss map. Hence \(\langle \varphi , \varphi \rangle _p=const.\) if and only if \(v=\pi ' \circ \overline{\varphi }=0\). In other words, an immersed submanifold of \({\mathbb {R}^{}} ^m\) is in addition contained in a hyperquadric \({\mathbb {Q}}^{m1}_{p,\epsilon r}\) if and only if its affine normal spaces (in \({\mathbb {R}^{}} ^m\)) are actually vectorial.
Proof
Lemma 4
If \(\overline{\varphi }: \mathcal{M}\rightarrow {\mathcal {Q}}\) is a map satisfying \(v=\pi ' \circ \overline{\varphi }=0\), then \(\beta \) vanishes. It follows that s must be parallel with respect to \(\nabla ^{N}.\)
We assume now that \(k=2\): let \(\overline{\varphi }: \mathcal{M}\rightarrow {\mathcal {Q}}_o=G(n,n+2)\). We write \(\overline{\varphi }= (e_1\wedge e_2)^\perp \), where \((e_1,e_2)\) is an orthonormal frame of \(\overline{\varphi }^\perp \) with \(\langle e_1,e_1\rangle _p=1\) and \(\langle e_2,e_2\rangle _p=\epsilon ,\)\(\epsilon =\pm 1.\) Hence if \(\epsilon =1\) the plane \(e_1 \wedge e_2\) has positive definite metric, and if \(\epsilon =1\) it has indefinite metric.
If \(\varphi =s\) is a section of \(E_N\) with \(\langle s, s\rangle =1\), there exists \(\theta \in C(\mathcal{M})\) such that \(\varphi = {\texttt {cos}\epsilon }(\theta ) e_1 + {\texttt {sin}\epsilon }(\theta ) e_2\), where \(({\texttt {cos}\epsilon },{\texttt {sin}\epsilon })=(\cos ,\sin )\) (resp. \((\cosh ,\sinh )\)) if \(\epsilon =1\) (resp. \(\epsilon =1\)). Observe that a unit normal vector along \(\varphi \) is given \(N:=\epsilon \, {\texttt {sin}\epsilon }(\theta ) e_1 + {\texttt {cos}\epsilon }(\theta ) e_2\).
Remark 10
This result may be also obtained as follows: we first note that the natural symplectic structure on \({\mathcal {Q}}_o\simeq G(2,n+2)\) may be interpreted as the curvature form \(\omega _o\in \Omega ^2({\mathcal {Q}}_o,End(\tau _N))\) of \(\tau _N\rightarrow {\mathcal {Q}}_o:\) this curvature form is indeed a 2form with values in the skewsymmetric operators acting on \(\tau _N,\) and may thus be naturally identified to a real form (since here the rank of \(\tau _N\) is 2). Finally, the existence of a nontrivial parallel section of \(\overline{\varphi }_o^{*}\tau _N\rightarrow \mathcal{M}\) is obviously equivalent to the vanishing of the curvature \(\overline{\varphi }_o^{*}\omega _o,\) since the rank of the bundle is 2.
Theorem 3
Let us consider \(\overline{\varphi }_o:\mathcal{M}\rightarrow G(2,n+2),\) a nparameter family of geodesic circles of \({\mathbb {S}^{}} ^{n+1}.\) We moreover assume that it is an immersion. There exists a hypersurface of \({\mathbb {S}^{}} ^{n+1}\) orthogonal to the family \(\overline{\varphi }_o\) if and only if the Lagrangian condition \(\overline{\varphi }_o^{*}\omega _o=0\) holds. When this is the case, there is a oneparameter family of such integral hypersurfaces which form a parallel family (with at most n singular leaves).
Remark 11
It appears here that condition (7) is not necessary: it may be proved that it is indeed a consequence of the other hypotheses (immersions of codimension 1 in space forms).
6.3 Submanifolds with Flat Normal Bundle in Space Forms
6.4 Surfaces with Nonvanishing Normal Curvature in 4Dimensional Space Forms
7 The Curvature of a Congruence
The aim of this section is to introduce the curvatures of a general family of affine kspaces in \({\mathbb {R}^{}} ^m.\) We begin with a formula expressing the curvatures of a submanifold in terms of its Gauss map, we then propose a general definition, and we finally establish a Gauss–Bonnet type formula for a 2parameter family of affine planes in \({\mathbb {R}^{}} ^4.\) The results of this section generalize results in Guilfoyle and Klingenberg (2004) concerning a general congruence of lines in \({\mathbb {R}^{}} ^3.\)
7.1 The Curvature of a Submanifold in Terms of its Gauss Map
Theorem 4
Remark 12
If \(\varphi :\mathcal{M}\rightarrow {\mathbb {R}^{}} ^m\) is an immersion with affine Gauss map \(\overline{\varphi },\) then the bundles \(E_T\rightarrow \mathcal{M}\) and \(E_N\rightarrow \mathcal{M}\) naturally identify with the tangent and the normal bundle of \(\mathcal{S}:=\varphi (\mathcal{M})\) respectively (we should say, with the pullbacks of these bundles on \(\mathcal{M}\)); moreover, under these identifications and by (4), the connections \(\nabla ^{T}\) and \(\nabla ^{N}\) identify with the Levi–Civita and the normal connections of \(\mathcal{S}.\) In particular, the Gauss and the normal curvature tensors of \(\mathcal{S}\) naturally identify with the curvature tensors of \(\nabla ^{T}\) and \(\nabla ^{N}\) respectively; thus, these tensors identify with the pullbacks of the curvature tensors of the tautological bundles \(\tau _T\rightarrow {\mathcal {Q}}_o\) and \(\tau _N\rightarrow {\mathcal {Q}}_o\) by the Gauss map \(\overline{\varphi }_o:\mathcal{M}\rightarrow {\mathcal {Q}}_o.\)
7.2 Generalized Curvature Tensor of a Congruence
Definition 2

a curvature tensor \(R:\ \Lambda ^2T_x\mathcal{M}\rightarrow \Lambda ^2\overline{\varphi }_o(x)\oplus \Lambda ^2\overline{\varphi }_o(x)^{\perp };\)

metrics on \(\overline{\varphi }_o(x),\)\(\overline{\varphi }_o(x)^{\perp }\) and \(T_x\mathcal{M}.\)
7.3 A Gauss–Bonnet Formula for a Congruence of Planes in \({\mathbb {R}^{}} ^4\)
Theorem 5
Remark 13
Proof
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