Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 77-81.

On Indicatrices^{1}

Sheila Carter F. J. Craveiro de Carvalho

School of Mathematics, Leeds University Leeds LS2 9JT, England e-mail: pmt6sc@amsta.leeds.ac.uk

Departamento de Matemática, Universidade de Coimbra Coimbra, Portugal e-mail: fjcc@mat.uc.pt

1. Introduction

We start by considering the space H of half-lines in R^{n+1} which can be identified with S^{n} × R^{n+1} in a natural way. Let f : M → R^{n+1} be a smooth, bounded map and assume that g : M → H is a smooth map such that, for x ∈ M, g(x) is a half-line starting at f(x). Now surround f(M) by a round n-sphere S and define F : M → S by taking F (x) to be the intersection point of g(x) with S. We will say that F is a spherical indicatrix. The purpose of the present note is to study the map F in some particular cases. Maps of this type have cropped up in some of our work [3], [4].

2. Normal indicatrices of a codimension 1 immersion

In what follows M will denote a boundaryless, compact, connected, oriented, n-dimensional manifold. Let f : M → R^{n+1} be a smooth immersion, smooth here meaning C^{∞}. Assume that S is a round n-sphere surrounding f(M). For what follows there is no loss of generality in assuming that the sphere is centred at the origin. Let now g : M → S^{n} × R^{n+1} be of the form (U, f). The indicatrix F_{U} is defined as follows. For x ∈ M, consider the half-line f(x) + αU(x), α ≥ 0. Then F_{U}(x) is the intersection of the half-line with S. We can write F_{U} = f + λU, where λ is a positive, smooth map.

Let us now denote by N : M → R^{n+1} the normal unit vector field determined in the following way. If, for x ∈ M, θ_{x} is the orientation for the tangent space T_{x}M, then [f_{∗x}(θ_{x}), N(x)] is the usual orientation of R^{n+1}. Here f_{∗x} denotes the induced linear map and the tangent space to R^{n+1} at x will be identified with R^{n+1} itself. The maps F_{N} and F_{−N} are the normal indicatrices.

1This is an original research article and no version has submitted for publication elsewhere.

Maps of the type f_{ξ} : M → R^{n+k}, with f_{ξ}(x) = f(x) + ξ(x), where ξ is a parallel normal field where studied by Carter and Şentürk [1] among other people.

If M is not diffeomorphic to S^{n} then any indicatrix will have critical points. We shall next characterise the critical points of the normal indicatrices.

Proposition 1. x is a critical point of F_{N} iff F_{N}(x) is a focal point of f with x as base point.

Proof. Let F_{N} = f + λN and x ∈ M. Use a chart ϕ at x and conclude that F_{N∗x} is injective iff the matrix I + λ(x)A is nonsingular, where I is the identity and A = [a_{ij}] is given by

N_{∗x}((\frac{∂}{∂ϕ_{i}})_{x}) = ∑_{j=1}^{n} a_{ji} f_{∗x}((\frac{∂}{∂ϕ_{j}})_{x}).

Next it can be shown that F_{N}(x) is a focal point of f with x as base point iff I + λ(x)A is singular. This can be done using the square-distance function L_{z}(y) =∥ f(y) − z ∥^{2}, with z = F_{N}(x), of which x is a critical point and the characterization of critical point degeneracy in terms of focal points as given in [6]. □

If f : S^{n} → R^{n+1} is an smooth immersion such that its Gaussian curvature does not vanish then one of the maps F_{N},F_{−N} is an immersion. Since nonvanishing Gaussian curvature is equivalent to focal set boundedness it is clear that if the radius of S is sufficiently large then both maps are immersions. A simple example where one of the normal indicatrices is not an immersion is shown below.

3. Degree of an indicatrix

Let us have the same assumptions and notations as in Section 2. We start with a result on mod 2 degrees.

Proposition 1. Let f : M → R^{n+1} be an immersion. Then degree_{2} F_{N} + degree_{2} F_{−N} ≡ e(M) mod 2, where e(M) stands for the Euler number of M.

Proof. Choose p ∈ S such that p is a regular value for both F_{N} and F_{−N}. Consider L_{p} : M → R given by L_{p}(x) =∥ f(x) − p ∥^{2}. The result follows easily from the fact that L_{p} is a Morse function [6] and the number of its critical points is congruent with e(M) mod 2. □

Proposition 2. Let M be even-dimensional. If F_{U} is an indicatrix such that, for x ∈ M, (U(x), N(x)) ≤ \frac{π}{2} then degree F_{U} = \frac{1}{2} e(M).

Proof. This follows from an old result of Heinz Hopf [5] and the fact that there is a homotopy between F_{U} and the map rN, where r is the radius of S. In fact, for t ∈ [0, 1], x ∈ M, H(x, t) = rN(x)+t(F_{U}(x)−rN(x)) is different from zero. If, for some t ≠ 0, 1, x ∈ M, H(t, x) were zero we would have F_{U}(x) = −rN(x) and consequently f(x) = −λ(x)U(x) − rN(x). Since ^{}(U(x), N(x)) ≤ \frac{π}{2}, that would imply ∥ f(x) ∥> r. We can then use H to define a homotopy between F_{U} and the map rN. □

If M is odd-dimensional there is still a homotopy but the result is no longer true. For instance, for M = S^{n}, odd n, we can have arbitrary odd degree. We refer the reader to the results in [5].

Assume now that M = S^{1} and that we are immersing it into R^{2}.

Proposition 3. Let F_{U} : S^{1} → S be an indicatrix such that, for s ∈ S^{1}, ^{}(U(s), N(s)) < π. Then F_{U} is homotopic to F_{N}.

Proof. Let T̃ : S^{1} × [0, 1] → S^{1} be such that T̃(s, t) = \frac{U(s)+t(N(s)−U(s))}{∥U(s)+t(N(s)−U(s))∥}. Use now T̃ to obtain H : S^{1} × [0, 1] → S given by H(s, t) = f(s) + μ(s, t)T̃(s, t), where μ(s, t) is obtained after finding the intersection of the half-line f(s) + αT̃(s, t), α ≥ 0, with S. □

We see that the tangential indicatrix, F_{T}, with T (x) the tangent vector to the curve at x, is homotopic to F_{N}. Using a rotation of angle πt in R^{2} we can show that F_{T} and F_{N} are homotopic to F_{−T} and F_{−N} respectively.

Let f : S^{1} → R^{2} be an immersion and p ∈ R^{2} \ f(S^{1}). We recall the following definitions. The rotation number of f is 2π degree T, T (x) being the tangent vector to f at x. On the other hand the winding number of f with respect to p is the degree of the map from S^{1} into itself given by \frac{f(x)−p}{∥f(x)−p∥}.

In the next two propositions both circles S^{1} and S are to be oriented in a similar way.

Proposition 4. Let f : S^{1} → R^{2} be such that no tangent line passes through 0. Then, for U as in Proposition 3, degree F_{U} is the winding number of f with respect to O.

Proof. It is enough to consider the case U = T. Now define H : S^{1} × [0, 1] → S by H(x, t) = r\frac{f(x)+tλ(x)T (x)}{∥f(x)+tλ(x)T (x)∥}, where r is the radius of S. □

Proposition 5. Let f : S^{1} → R^{2} be such that its curvature does not vanish. Then, for U as in Proposition 3, 2π degree F_{U} = rot f, where rot f stands for the rotation number of f.

Proof. Again we consider he case U = T. From F_{T} = f + λT it is clear that f and F_{T} are regularly homotopic and consequently rot f = rot F_{T}. Since rot F_{T} = 2π degree F_{T} the result follows. □

4. Applications

There is no reason to consider just immersions with codimension 1. An interesting situation occurs with curves in 3-space.

A. Curves with small total torsion

In [2] it was convenient at some stage to indicate how curves with small total torsion could be obtained. There we used a convenient non-degenerate homotopy as suggested by [7]. Here we will use another type of homotopy for a similar purpose.

Let f : S^{1} → R^{3} be a closed curve with nonvanishing torsion. Consider F_{T} : S^{1} → S given by F_{T} (x) = f(x)+λ(x)T (x), where T (x) is the unit tangent vector at x. For 0 ≤ t ≤ 1, g_{t} = f + tλT gives rise to a non-degenerate homotopy, that is one that at every stage t the corresponding curve g_{t} has curvature which vanishes nowhere. Since under a non-degenerate deformation the total torsion varies continuously and the total torsion of a spherical curve is zero it follows that curves with very small, nonzero total torsion can be obtained.

B. Linking numbers

Let f, g : S^{1} → R^{3} be curves with disjoint images. The linking number L(f, g) is the degree of the map ϕ : S^{1} × S^{1} → S^{2} given by ϕ(x, y) = \frac{f(x)−g(y)}{∥f(x)−g(y)∥}.

Proposition 1. Let f, g : S^{1} → R^{3} be curves such that the image of one of them does not intersect any tangent line to the other. Then the linking number L(f, g) is zero.

Proof. Assume that no tangent line to f meets g(S^{1}). Consider F_{T} = f + λT. Then f is homotopic to F_{T} and the homotopy induces a homotopy between ϕ : S^{1} × S^{1} → S^{2}, given by ϕ(x, y) = \frac{f(x)−g(y)}{∥f(x)−g(y)∥} and ψ : S^{1} × S^{1} → S^{2}, given by ψ(x, y) = \frac{F_{T} (x)−g(y)}{∥F_{T} (x)−g(y)∥}. Therefore L(f, g) = L(F_{T} , g). If we choose the 2-sphere S for the definition of F_{T} sufficiently big it follows that L(F_{T} , g) = 0. □

Let us recall that two space curves f and g are athwart if no tangent line to f intersects a tangent line to g [4]. Athwart curves are examples of curves in the conditions of Proposition 1. It is known [4] that there are curves which cannot be athwart to any other curve. We are going to show that on the other hand given a curve we can always find another one such that the conditions of Proposition 1 are satisfied.

Proposition 2. Let f : S^{1} → R^{3} be a curve. Then, for every sphere containing f(S^{1}) in its interior, there is a spherical curve g : S^{1} → R^{3} such that no tangent line to f meets g(S^{1}).

Proof. We follow [3] where we showed that the tangent lines to f do not fill R^{3}. Consider F_{T} , F_{−T}. Then X = F_{T} (S^{1}) ∪ F_{−T} (S^{1}) is a set of which the complement in S^{2} is open and nonempty. Any curve g : S^{1} → R^{3} with image in S^{2} \ X will do. □

In the statement above spherical may be replaced by plane as we show next.

Proposition 2′. Let f : S^{1} → R^{3} be a curve. Then, for every plane which does not meet f(S^{1}), there is a plane curve g : S^{1} → R^{3} such that no tangent line to f meets g(S^{1}).

Proof. Take a plane π such that f(S^{1}) ∩ π = ∅. Consider the projective closure RP^{2} of π and define F : S^{1} → RP^{2} by letting F (x) be the intersection point of the tangent line to f at x with π. Such a map F is smooth and again by Sard’s Theorem it is possible to find a non-empty open subset of π which avoids F (S^{1}). Take for g any curve with image in that open subset. □

Obviously similar results can be obtained replacing tangent by principal normal or binormal if the extra assumption of nonvanishing curvature is imposed where necessary.

References

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

[2] Craveiro de Carvalho, F. J.; Robertson, S. A.: Self-parallel curves. Math. Scand. 65 (1989).

[3] Craveiro de Carvalho, F. J.; Robertson, S. A.: A remark on cylindrical curves and induced curves in RP^{2}. Note di Matematica 16 (1996)(1).

[4] Craveiro de Carvalho, F. J.; West, A.: Athwart curves in R^{3}. J. Geom. 26 (1986), 82–98.

[5] Milnor, J.: On the immersion of n-manifolds in (n + 1)-space. Comment. Math. Helv. 30 (1956), 275–284.

[6] Milnor, J.: Morse Theory. Princeton University Press 1973.

[7] Pohl, W. F.: The self-linking number of a closed space curve. J. Math. Mech. 17 (1968), 975–985.