Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 1-6.

Parallels and Push-outs of Immersed Manifolds

Sheila Carter

School of Mathematics, University of Leeds, Leeds LS2 9JT, England e-mail: s.carter@leeds.ac.uk

1. Introduction

It is a straightforward calculation to show that if f : M^{m} → R^{m+1} is a smooth immersion of a connected, smooth manifold and ξ : M^{m} → R^{m+1} is a normal field for f with constant length, then f + ξ : M → R^{m+1} is an immersion if and only if, for all p ∈ M, (f + ξ)(p) is not a focal point of f with base p. Such an immersion f + ξ is then called a parallel immersion of f. Here I am going to survey results on parallel immersions and their generalizations.

2. Basic Definitions and Examples

Throughout this paper M^{m} will denote a boundaryless, connected, smooth (C^{∞}) m- dimensional manifold and f : M^{m} → R^{m+k} will be a smooth immersion of M into Euclidean (m + k)-space.

The total space of the normal bundle of f is

N(f) = {(p, x) ∈ M × R^{m+k} : x ∈ (f_{∗}T_{p}M)^{⊥}}

and the endpoint map η : N(f) → R^{m+k} is defined by η(p, x) = f(p) + x. The set of singularities of η will be denoted by Σ(f). The focal set of f is then η(Σ(f)) ⊂ R^{m+k}.

At p ∈ M, let N_{p}(f) = {x ∈ R^{m+k} : (p, x) ∈ N(f)} and Σ_{p}(f) = {x ∈ R^{m+k} : (p, x) ∈ Σ(f)}. Then Σ_{p}(f) ⊂ N_{p}(f) can be thought of as the set of focal points of f with base p, although, strictly speaking, it is the subset f(p)+Σ_{p}(f) of the affine normal plane f(p)+N_{p}(f) which is the set of focal points with base p.

As in [8] the holonomy group of the normal bundle is used to compare normal planes and the focal sets in these planes.

Fix p_{0} ∈ M and for p ∈ M and a (piecewise smooth) path γ from p_{0} to p let φ_{p,γ} : N_{p_{0}}(f) → N_{p}(f) be the isometry defined by parallel transport in N(f) along γ using the normal connection.

The normal holonomy group on N_{p_{0}}(f) is

Hol(f) = {φ_{p_{0},γ} : γ is a path from p_{0} to p_{0}}.

By taking an isometry from N_{p_{0}}(f) to R^{k}, Hol(f) can be thought of as a subgroup of O(k) acting on R^{k}. Crucial to this work are the results on the normal holonomy group by C.Olmos [14].

The push-out space of f

Ω(f) = {x ∈ N_{p_{0}}(f) : ∀p ∈ M and ∀ paths γ from p_{0} to p, φ_{p,γ}(x) ∉ Σ_{p}(f)}.

So Ω(f) is invariant under the action of Hol(f). It is the set of normals at p_{0} which, when transported parallelly along all curves from p_{0}, do not meet focal points.

We shall see that each point of Ω(f) determines an immersion of some bundle into R^{m+k}. For x ∈ N_{p_{0}}, let W_{x} ⊂ N(f) be defined by

W_{x} = {(p, φ_{p,γ}(x)) : p ∈ M, γ is a path from p_{0} to p}.

It is a subbundle of the normal bundle and its fibre is the orbit of Hol(f) through x. In [12] W_{x} is called the holonomy subbundle of N(f) through x.

If Hol(f) is closed in O(k) (that is, Hol(f) is compact) then W_{x} is a submanifold of N(f). If it is not closed then this difficulty can be avoided by replacing M by its simply connected covering space M̃ and replacing f by f̃ = f ∘ π : M̃ → R^{m+k}, where π : M̃ → M is the covering projection. Then Ω(f̃) is isometric to Ω(f) and Hol(f̃) is compact [8]. Now η | W_{x} : W_{x} → R^{m+k} is an immersion

⟺ W_{x} ∩ Σ(f) = ∅
⟺ x ∈ Ω(f).

If x ∈ Ω(f) then η | W_{x} : W_{x} → R^{m+k} is called a push-out of f. We shall see that the dimensions of the submanifolds W_{x} can vary with x.

In the special case where Hol(f) is trivial, there exists an orthonormal set of parallel normal fields n_{1}, …, n_{k} : M → R^{m+k}. Then, if x ∈ N_{p_{0}}(f), x = ∑_{i=1}^{k} x_{i}n_{i}(p_{0}) for some x_{1}, …, x_{k} ∈ R. Put ξ = ∑_{i=1}^{k} x_{i}n_{i} : M → R^{m+k}. Then ξ is a parallel normal field and for any path γ from p_{0} to p ∈ M, φ_{p,γ}(x) = ξ(p). Thus in this special case, W_{x} = {(p, ξ(p)) : p ∈ M} which is a section of N(f) and can be identified with M, and (η | W_{x})(p, ξ(p)) = (f + ξ)(p). In this case, if x ∈ Ω(f) we say that η | W_{x} is a parallel of f. So a parallel is a special case of a push-out. Whereas parallels of f : M → R^{m+k} are immersions of M, a push-out can be an immersion of a manifold with dimension greater than m. The push-out space of an immersion was studied in [4] for the special case in which Hol(f) is trivial, and in [8] for the general case. Other general accounts and recent results can be found in [1, 2, 3, 5, 6].

If Σ(f) is invariant under parallel transport (that is, Σ_{p_{0}}(f) is invariant under the action of the normal holonomy group) then Ω(f) is isometric to N_{p_{0}}(f) \ Σ_{p_{0}}(f). This situation can only happen if f(M) is an isoparametric submanifold or a focal manifold of an isoparametric submanifold, that is, f(M) has constant principal curvatures [12]. A description of a Veronese surface for which the push-out space has three path-connected components is described in [6]. Some of the push-outs are Cartan’s 3-dimensional isoparametric submanifold.

Examples of the push-out space of immersed curves can be found in [1, 8].

Let K^{2} denote the Klein bottle. In [3] examples are given of an immersion f : K^{2} → R^{4} and an embedding g : K^{2} → R^{5} for which Ω(f) and Ω(g) have four path-connected components. It is not known whether or not there exists an embedding of K^{2} in R^{4} with this property.

3. Self-parallel immersions

In [10], H.Farran and S.A.Robertson defined two immersions f, g : M^{m} → R^{m+k} to be parallel if, for all p ∈ M, f(p) + N_{p}(f) = g(p) + N_{p}(g). A diffeomorphism δ : M → M is called a self-parallelism of M with respect to f if f and δ ∘ f are parallel. They studied the group of all self-parallelisms of M with respect to f.

It was shown by A.M. Flegmann [11] and B. Wegner [17] that f and g are parallel if and only if g = f + ξ where ξ is a parallel normal field for f. So g is a parallel of f in the sense of Section 2. The parallel rank, ρ(f), of an immersion f : M^{m} → R^{m+k} is defined to be the dimension of the affine space spanned by the immersions which are parallels of f. Thus 0 ≤ ρ(f) ≤ k and ρ(f) = k if and only if Hol(f) is trivial. For results on parallel rank and examples illustrating related concepts see [9, 10, 17]. Self-parallel curves are studied in [9, 13, 18]. Effectively, a self-parallel curve is a push-out from a central curve [18].

Some of the above ideas are extended in [21] to parallel curves on surfaces, in [22] to parallel immersions into spaces of constant curvature, and in [19, 20] to parallelism for polygons and polyhedra. A general account of parallelism for both smooth curves and polygons can be found in [23] which includes two animated examples on self-parallel polygons.

4. Theorems on the push-out space

As usual, f : M^{m} → R^{m+k} denotes an immersion and Ω(f) denotes the push-out space of f. The following theorems are about the number of path-connected components of Ω(f) and the topology of Ω(f).

Theorem 1. [8] Ω(f) has only a finite number of path-connected components.

Theorem 2. [4, 6, 8, 16] Let M be compact and let k = 1, then

(1) if M is orientable, Ω(f) has at most two components,

(2) if M is nonorientable, Ω(f) is connected.

Essentially the first part of this theorem was proved by B. Smyth who considered the connected components of the space

{λ ∈ R : for some p ∈ M, λ is a principal curvature at p}.

He also considered the corresponding problem for immersions f : M^{m} → S^{m+1} [16]. It is not known whether or not there exists a complete embedding f : M^{m} → R^{m+1} of a noncompact manifold for which Ω(f) has more than two components.

Theorem 3. [8] Let M be compact then Ω(f) is a neighbourhood of the origin in N_{p_{0}}(f).

An example is given in [8] in which Ω(f) is not open. However, it is shown in [4] that if M is compact and Hol(f) is trivial then Ω(f) is open.

Theorem 4. [8] The path-connected component of Ω(f) to which 0 belongs is star-like.

It was shown in [4] that if Hol(f) is trivial then the path-connected components of Hol(f) are convex. This can be generalised by considering Hol_{0}(f), the component of the identity in Hol(f), that is

Hol_{0}(f) = {φ_{p_{0},γ} ∈ Hol(f) : γ is homotopically trivial }.

It is a consequence of the work of C.Olmos [14] that through each x ∈ N_{p_{0}}(f) there is an affine plane Π ⊂ N_{p_{0}}(f) which intersects all the orbits of Hol_{0}(f) orthogonally and is an affine normal plane to the regular orbits. Using this comment Ω(f)/Hol_{0}(f) can be identified with a subset of a vector space.

Theorem 5. [8] Each connected component of Ω(f)/Hol_{0}(f) is convex.

Theorem 6. [8] Let M be compact and let Hol(f) act transitively on S^{k−1} ⊂ R^{k} ≡ N_{p_{0}}(f). Then Ω(f) is an open ball in R^{k}.

The second part of Theorem 2 is a special case of this since M being nonorientable is equivalent to Hol(f) being transitive on S^{0} = {−1, 1}.

Theorem 7. [8] Let Hol(f) have an orbit that is dense in S^{k−1} but Hol_{0}(f) is not transitive on S^{k−1} ⊂ R^{k} ≡ N_{p_{0}}(f). Then, either Ω(f) = {0} or the interior of Ω(f) is an open ball in R^{k} whose closure contains Ω(f).

Examples illustrating these theorems can be found in [8].

There are still many interesting problems to be solved in the case in which Hol(f) is trivial. In this case the focal set in each normal plane consists of at most m hyperplanes (where m = dim M). Put

α(m, k) = \{ \begin{matrix}2^{m} & for m ≤ k \\ Σ_{r=0}^{k}\binom{m}{r} & for m > k \end{matrix}

and

γ(m, k) = \{ \begin{matrix}2^{m} & for m ≤ k \\ 2Σ_{r=0}^{k−1}\binom{m−1}{r} & for m > k \end{matrix}

so that α(m, k) (resp. γ(m, k)) is the number of components (resp. unbounded components) in the complement of m hyperplanes in general position in R^{k}. Let d(f) denote the number of path-connected components of Ω(f).

Theorem 8. [4] Let Hol(f) be trivial then d(f) ≤ α(m, k).

Conjecture. Let M be compact and let Hol(f) be trivial then d(f) ≤ γ(m, k).

This is trivially true if m ≤ k since then γ(m, k) = α(m, k). The first part of Theorem 2 shows that the conjecture is true for k = 1, as γ(m, 1) = 2. Also the conjecture has been proved recently in the case k = 2 where γ(m, 2) = 2m [15]. For all m ≥ 1, k ≥ 1 it has been shown that there exists an embedding f : S^{1} × … × S^{1} → R^{m+k} of the product of m copies of S^{1}, for which d(f) = γ(m, k), [1, 7]. In order to get large d(f) the manifold seems to need a large number of generators for its homology.

Evidence which supports the conjecture is the unboundedness property.

Theorem 9. [4] Let M be compact, let Hol(f) be trivial and let A be a path-connected component of Ω(f) ⊂ N_{p_{0}}. Then there exists p ∈ M such that the path-connected component of N_{p}(f) \ Σ_{p}(f) which contains the set {φ_{p,γ}(A) : γ is a path from p_{0} to p} is unbounded.

More information can be obtained about Ω(f) by assigning an integer to each of its pathconnected components, called the index of the component. Roughly, if x ∈ Ω(f) ⊂ N_{p_{0}}, the index of x is the number of focal points of f with base p_{0} (counted with a multiplicity) on the line segment from x to p_{0}. This number is constant over a path-connected component of Ω(f). The possible indices that can occur are related to the topology of the manifold.

Theorem 10. [2] Let f : M^{m} → R^{m+k} be an immersion of a compact manifold and let Hol(f) be trivial then if Ω(f) has a component with odd index, the Euler characteristic of M is zero.

This was proved for the case k = 1 in [4] where there are also results relating the index and the ranks of the homology groups of M. See also [5].

References

[1] S. Carter: Immersions parallel to a given immersion. Geometry and Topology of Submanifolds V (ed. F. Dillen, L. Verstraelen, L. Vrancken, I. van de Woestijne), World Scientific, Singapore 1993, 68–69.

[2] S. Carter and Y. Kaya: Immersions with a parallel normal field. University of Leeds, Dept. of Pure Maths. 1999 Preprint Series, No. 24.

[3] S. Carter and Y. Kaya: The push-out space of spherically immersed surfaces. University of Leeds, Dept. of Pure Math. 1999 Preprint Series, No. 27.

[4] S. Carter and Z. Şentürk: The space of immersions parallel to a given immersion. J. London Math. Soc. (2) 50 (1994), 404–416.

[5] S. Carter and Z. Şentürk: On the index of parallel immersions. Geometry and Topology of Submanifolds VI (ed. F. Dillen, I. Van de Wostijne, L. Verstraelen), World Scientific, Singapore 1994, 43–49.

[6] S. Carter, Z. Şentürk and A. West: The pushout space of a submanifold. Geometry and Topology of Submanifolds VI (ed. F.Dillen, I. Van de Woestijne, L.Verstraelen), World Scientific, Singapore 1994, 50–57.

[7] S. Carter and A. West: Partial tubes about immersed manifolds. Geom. Dedicata 54 (1995), 145–169.

[8] S. Carter and A. West: The push-out region of an immersed manifold. University of Leeds, Dept. of Pure Math. Preprint Series 1998, No. 29.

[9] F. J. Craveiro de Carvalho and S. A. Robertson: Self-parallel curves. Math. Scand. 65 (1989), 67–74.

[10] H. R. Farran and S. A. Robertson: Parallel immersions in Euclidean space. J. London Math. Soc. (2) 35 (1987), 527–538.

[11] A. M. Flegmann: The parallel rank of a submanifold of Euclidean space. Math. Proc. Camb. Phil. Soc. 106 (1989), 89–93.

[12] E. Heintze, C. Olmos and G. Thorbergsson: Submanifolds with constant principal curvatures and normal holonomy groups. Internat. J. Math., Vol. 2, No. 2 (1991), 167–175.

[13] T. F. Mersal and B. Wegner: Variation of the total normal twist of closed curves in Euclidean spaces. Proceedings 1st International Meeting on Geometry and Topology, (Braga 1997), Cent. Mat. Univ. Minho, Braga 1998, 223–231.

[14] C. Olmos: The normal holonomy group. Proc. Amer. Math. Soc. 110 (1990), 813–818.

[15] I. Silva and A. West: Unbounded regions in an arrangement of lines in the plane. In preparation.

[16] B. Smyth: Efimov’s inequality and other inequalities in a sphere. Geometry and Topology of Submanifolds IV (eds. F. Dillen and L. Verstraelen), World Scientific, Singapore 1992, 76–86.

[17] B. Wegner: Some remarks on parallel immersions. Diff. Geom. Appl. (Eger 1989), Coll. Math. Soc. J. Bolyai, 56, North-Holland, Amsterdam 1992, 707–717.

[18] B. Wegner: Self-parallel and transnormal curves. Geom. Dedicata 38 (1991), 175–191.

[19] B. Wegner: Exterior parallelism for polyhedra. Math. Pannon. 2 (1991), 95–106.

[20] B. Wegner: Self-parallel polygons and polyhedra. Proc. Third Congress Geom. (Thessaloniki 1991), Aristotle Univ. Thessaloniki 1992, 444–451.

[21] B. Wegner: Parallel and transnormal curves on surfaces. New Developments in Diff. Geom. (ed. L. Tamássy, J. Szenthe), Proc. Coll. Diff. Geom. (Debrecen 1994), Kluwer Acad. Publ., Dordrecht, Math. Appl. 350 (1996), 423–434.

[22] B. Wegner: Parallel immersions into spaces of constant curvature and conformal transformations. Proc. Third Int. Workshop Diff. Geom. Appl. and First German-Romanian Seminar on Geom. (Sibiu, Romania 1997), General Mathematics 5 (1997), 401–407.

[23] B. Wegner: Tangential symmetries of planar curves and space curves. Visual Mathematics 1, (http://www.mi.sanu.ac.yu/vismath/wegner/index.html).