The Minimum Number of Zeros of Lipschitz-Killing Curvature

Dorin Andrica and Cornel Pintea

Abstract

In the previous paper [11] some sufficient topological conditions, expressed in terms of homotopy groups, of two given differentiable manifolds M, N in order that the φ-category of the pair (M, N) be infinite, are given. In the more recent paper [12] it is proved that, in the same topological conditions, the φ-category is actually infinite uncountable. As an application we observe in this paper that a lower bound for the minimum number of points of zeros of the Lipschitz-Killing curvature of some m-dimensional manifold immersible in R^{m+k}, with respect to all this kind of immersions, can be given in terms of φ-category.

Mathematics Subject Classification: 53Q99, 53Q05, 55P99, 53P65, 55Q05 Key words: Lipschitz-Killing curvature, homotopy group, critical set, immersiability category

1 Introduction

Recall that the φ-category of the pair (M, N) of closed manifolds is defined as

φ(M, N) = min{#C(f) | f ∈ C^{∞}(M, N)},

where C(f) is the critical set of f. A survey containing general results concerning φ-category is contained in the paper [1] (see also [2] and [3]).

Let M^{m} be a differentiable manifold immersible in R^{m+k}; let f : M → R^{m+k} be an immersion and

v ∈ (df)_{p}(T_{p}(M))^{⊥} ∩ S^{m+k−1},

where p ∈ M is a given point. Consider the second fundamental form ϕ_{f,p}^{v} of f and the projections N_{f} : N^{f,1} → S^{m+k−1}, π : N^{f,1} → M, where

N^{f,1} = {(x, v) ∈ M × S^{m+k−1} : v ⊥ (df)_{p}(T_{p}(M))} =

= ⋃_{p∈M} ({p} × (S^{m+k−1} ∩ (df)_{p}(T_{p}(M))^{⊥})).

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 1-6.

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Observe that π is a fibration with fibre S^{k−1}. Recall that the Lipschitz-Killing curvature of f at the point p ∈ M in the direction v is defined as

L_{f,v} = \frac{det(ϕ_{f,p}^{v})}{det(g_{f}(p))}

and that ϕ_{f,p_{0}}^{v_{0}} is degenerated if and only if (p_{0}, v_{0}) is a critical point of N_{f} ( for more details we refer to the books [4] and [6]). Therefore

C(N_{f}) = {(p, v) ∈ N^{f,1} : L_{f,v}(p) = 0},

that is the φ-category of the pair (N^{f,1}, S^{m+k−1}) is a lower bound for the minimum number of points of zeros of the Lipschitz-Killing curvature with respect to all immersions of M in R^{m+k}. This fact can be shortly written by means of the inequality

L_{k}(M) ≥ φ(N^{f,1}, S^{m+k−1}),

where

L_{k}(M) = min{#C(N_{f}) : f ∈ Imm^{k}(M)}

is the so called L_{k}-category or the immersiability category of M, and Imm^{k}(M) is the set of all immersions of M into R^{m+k}. The immersiability category L_{k}(M) is studied in the case of surfaces and k = 1 in the recent second author’s paper [12]. To prove that the L_{k}(M) is infinite uncountable in certain situations, we will use the following theorem.

Theorem 1.1 ([12]) Let M, N be compact connected differentiable manifolds of the same dimension m.

(i) If m ≥ 3 and π_{1}(M) cannot be embedded as a subgroup in π_{1}(N), then φ(M, N) = ℵ_{1};

(ii) If m ≥ 4 and π_{q}(M) ≄ π_{q}(N) for some q ∈ {2, …, m−2}, then φ(M, N) = ℵ_{1}.

Let us mention that Theorem 1.1 is improving ([11], Theorem 3.1) which states that, under the same hypothesis, the φ-category of the pair (M, N) is just ≥ ℵ_{0}.

2 The implications of Theorem 1.1 on the L_{k}-category

Because the conditions of Theorem 1.1 are expressed in terms of homotopy groups of the given manifolds and our lower bound for the minimum number of points of zeros of the Lipschitz-Killing curvature is φ(N^{f,1}, S^{m+k−1}), we will try to relate in what follows the homotopy groups of N^{f,1} with those of M. We denote by k(M) the smallest natural number such that M is immersible in R^{m+k(M)}. By means of Whitney’s theorem, observe that k(M) ≤ 2m + 1 and if M^{m} is a compact manifold, then k(M) ≤ 2m − 1. Using [9], Theorem 1 and the proof of [9], Theorem 2 one can see some lower bounds for k(G_{2,n}) and k(G_{3,n}). For some lower bounds of k(P^{n}(R)) and of k(P^{n}(C)), k(P^{n}(H)) see [8] and [7] respectively.

Theorem 2.1 Let M^{m} be a connected closed manifold.

(i) If m ≥ 1 and π_{1}(M) is not trivial, then L_{k}(M) = ℵ_{1}, ∀k ≥ max{3, k(M)};

(ii) If m ≥ 2 and π_{q}(M) is not trivial for some for some q ∈ {2, …, k − 1}, then L_{k}(M) ≥ ℵ_{1}, ∀k ≥ max{3, k(M)}.

Proof. The homotopy sequence of the fibration S^{k−1} ↪ N^{f,1} →^{π} M is

⋯ → π_{r}(S^{k−1}) → π_{r}(N^{f,1}) → π_{r}(M) → π_{r−1}(S^{k−1}) → · · · .

Because the homotopy groups π_{q}(S^{k−1}), π_{q−1}(S^{k−1}) of the sphere are trivial for q ∈ {1, …, k − 2} and π_{k−1}(S^{k−1}) ≃ Z it follows that π_{q}(N^{f,1}) ≃ π_{q}(M) for all q ∈ {1, …, k − 2} and the morphism π_{k−1}(N^{f,1}) → π_{k−1}(M), induced by the projection π : N^{f,1}) → M is surjective. Therefore, to prove (i) apply Theorem 1.1 (i), taking into accout that π_{1}(M) ≄ 0 ≃ π_{1}(S^{m+k−1}). Annalogously, to prove (ii) apply theorem 1.1 (ii), taking into accout that for some for some q ∈ {2, …, k − 1} we have π_{q}(M) ≄ 0 ≃ π_{q}(S^{m+k−1}).

Theorem 2.2 Let M^{m} be a connected closed manifold immersible in R^{m+2}.

(i) If m ≥ 2 and π_{1}(M) is not trivial, then L_{2}(M) = ℵ_{1};

(ii) If m ≥ 3 and π_{2}(M) ≄ Z, then L_{2}(M) = ℵ_{1};

(iii) If m ≥ 4 and π_{q}(M) is not trivial for some for some q ∈ {3, …, m − 1}, then L_{2}(M) = ℵ_{1}.

Proof. The homotopy sequence of the fibration S^{1} ↪ N^{f,1} →^{π} M is

⋯ → π_{r}(S^{1}) → π_{r}(N^{f,1}) → π_{r}(M) → π_{r−1}(S^{1}) → · · · .

Because the homotopy groups π_{q}(S^{1}) of the circle are trivial for q = 0 and q ≥ 2 and π_{1}(S^{1}) ≃ Z, it follows that the morphism π_{1}(N^{f,1}) → π_{1}(M) induced by the projection π : N^{f,1} → M is surjective, meaning that π_{1}(M) ≄ 0 implies π_{1}(N^{f,1}) ≄ 0 ≃ π_{1}(S^{m+1}). Therefore using Theorem 1.1 (i), it follows that L_{2}(M) = ℵ_{1} and the statement (i) is proved. To prove the second statement, observe that the morphism

π_{2}(N^{f,1}) → π_{2}(M)

induced by the projection π : N^{f,1} → M is injective and

π_{2}(M)/π_{2}(N^{f,1}) ≃ Z.

Combining this observation with the hypothesis of (ii) it follows that

π_{2}(N^{f,1}) ≠ 0 ≃ π_{2}(S^{m+1}).

Therefore using Theorem 1.1 (ii) conclude that L_{2}(M) = ℵ_{1}.

The statement (iii) follows immediately from Theorem 1.1 (ii) taking into account that for q ∈ {3, …, m − 1} we have π_{q}(N^{f,1}) ≃ π_{r}(M) ≄ 0 ≃ π_{q}(S^{m+1}).

3 Some manifolds with infinite uncountable L_{k}-category

In this section we are going to use the results of the previous section to obtain some manifolds with infinite L_{k}-category.

Proposition 3.1 (i) For m ≥ 1 we have that

L_{k}(P^{m}(R)) = ℵ_{1}, ∀k ≥ max{3, k(P^{m}(R))};

(ii) For m ≥ 1 we have that

L_{k}(L^{2m−1}(q_{1}, …, q_{p})) = ℵ_{1}, ∀k ≥ max{3, k(L^{2m−1}(q_{1}, …, q_{p}))};

(iii) For m ≥ 2 we have that L_{k}(SO_{m}) ≥ ℵ_{1}, ∀k ≥ max{3, k(SO_{m})}.

Proof. Indeed, for m ≥ 3 we have

π_{1}(P^{2}(R)) ≃ π_{1}(P^{m}(R)) ≃ π_{1}(SO_{m}) ≃ Z_{2} ≠ 0,

while P^{1}(R) ∼_{T op} SO_{2} ∼_{T op} S^{1} so that

π_{1}(P^{1}(R)) ≃ π_{1}(SO_{2}) ≃ Z ≠ 0,

and

π_{1}(L^{2m−1}(q_{1}, …, q_{p})) ≃ Z_{m} ≠ 0, ∀m ≥ 1.

Therefore using Theorem 2.1 (i), proposition 3.1 is completely proved.

Proposition 3.2 (i) L_{k}(G_{p,l}) = ℵ_{1}, ∀k ≥ max{3, k(G_{p,l})}; (ii) If m ≥ 3, then L_{k}(Spin_{m}) = ℵ_{1}, ∀k ≥ max{4, k(Spin_{m})}.

Proof. (i) Let us assume that p ≥ l. In this case it follows, using [13], Theorem 10.16, pp. 204, that

π_{q}(G_{p,l}) ≃ π_{q}(V_{p+l,l}) ⊕ π_{q−1}(O_{l}), ∀q.

On the other hand the inclusion SO_{l} ↪ O_{l} induces isomorphism of homotopy groups π_{i}(SO_{l}) → π_{i}(O_{l}), so that for l = 2 we have π_{1}(O_{2}) ≃ π_{1}(SO_{2}) ≃ Z and for l ≥ 3 we have π_{1}(O_{l}) ≃ π_{1}(SO_{l}) ≃ Z_{2}. In any case π_{2}(G_{p,l}) has as a subgroup either Z or Z_{2}, that is, π_{2}(G_{p,l}) ≄ 0, meaning by Theorem 2.1 (ii), that L_{k}(G_{p,l}) = ℵ_{1}, ∀k ≥ max{3, k(G_{p,l})}.

(ii) Indeed π_{q}(Spin_{m}) ≃ π_{q}(SO_{m}), ∀k ≥ 2, Spin_{m} being the universal cover of SO_{m}. Hence

π_{3}(Spin_{m}) ≃ π_{3}(SO_{m}) ≃ \{ \begin{matrix}Z & if m = 3 and m ≥ 5 \\ Z ⊕ Z & if m = 4. \end{matrix}

For the above mentioned homotopy groups, see for instance [5], pp. 224. Therefore in the given conditions, π_{3}(Spin_{m}) ≄ 0, meaning by Theorem 2.1 (ii), that L_{k}(Spin_{m}) = ℵ_{1}, ∀k ≥ max{4, k(Spin_{m})}.

Remark 3.3 For k ≥ max{3, p + 1, k(G_{p,l})} the first point of Proposition 3.2 can be also proved by means of [13], Theorem 10.13, pp. 203, from which it follows that π_{p}(V_{p+l,l}) is either infinit cyclic or cyclic of order two.

Acknowledgements. This paper is supported by the European Research and Training Network ”GEOMETRIC ANALYSIS”. It was written in a period when the second author had a research appointment position at ”Paul Sabatier” University, Toulouse, France (November 1, 2000-May 1, 2001)

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”Babeş-Bolyai” University, Faculty of Mathematics, Str. M. Kogălniceanu 1, 3400 Cluj-Napoca, Romania E-mail addreses: dandrica@math.ubbcluj.ro

cpintea@math.ubbcluj.ro