PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 83(97) (2008), 65–69 DOI: 10.2298/PIM0897065P

ON THE DIFFERENTIABILITY OF A DISTANCE FUNCTION

Kwang-Soon Park

Communicated by Stevan Pilipović

Abstract. Let M be a simply connected complete Kähler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M. Let U_{ν} be the unit normal bundle of N in M. We prove that if a distance function ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at −v.

1. Introduction

Let N be a closed submanifold of a complete Riemannian manifold M and π : U_{ν} ↦ N the unit normal bundle of N in M. For v ∈ T_{p}M, p ∈ M, throughout this paper, let γ_{v}(t) denote always the geodesic curve such that γ_{v}(0) = p and γ′_{v}(0) = v. Define a function ρ : U_{ν} ↦ ℝ by

ρ(v) := sup{t > 0 | d(N, γ_{v}(t)) = t} for v ∈ U_{ν},

where d(N, γ_{v}(t)) denotes the distance between N and γ_{v}(t). For each positive integer k ∈ ℕ, define a function λ_{k} : U_{ν} ↦ ℝ by

λ_{k}(v) := sup{t > 0 | γ_{v}|_{[0,t]} has no k-th focal point of N}

for v ∈ U_{ν} [2]. The followings are well known: ρ is continuous [10] and λ_{1} is smooth where λ_{1} is finite [2]. Itoh and Tanaka [2] proved that the function ρ on U_{ν} is locally Lipschitz, where ρ is finite. So, by Rademacher’s theorem ([1], [6]), the function min(ρ, r) is differentiable almost everywhere for each r > 0. Generally, it is well known that ρ is differentiable at v ∈ U_{ν} if γ_{v}(ρ(v)) is a normal cut point, i.e., there exist exactly two N-segments through γ_{v}(ρ(v)) such that γ_{v}(ρ(v)) is not a focal point along all of these two N-segments. Furthermore, in the case dim M = 2, Tanaka [8] proved that a point v ∈ U_{ν} with ρ(v) < ∞ is a differentiable point of the function ρ if and only if γ_{v}(ρ(v)) is a 1-st focal point of N along γ_{v} or there exist at most two N-segments through γ_{v}(ρ(v)). Here, a curve γ : [0, r] ↦ M is called an N-segment if γ is a geodesic curve such that γ′(0) ∈ U_{ν} and d(N, γ(t)) = t for t ∈ [0, r]. This fact is obviously very nice but didn’t have any information about the n-dimensional manifold M with n ⩾ 3. So, we plan to consider the manifold M such that it has some good conditions. Then we have

2000 Mathematics Subject Classification: Primary 53C22; 53C55.

Supported by the BK21 project of the Ministry of Education, Korea.

Main Theorem. Let M be a simply connected complete Kähler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M. Let U_{ν} be the unit normal bundle of N in M. If ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at −v.

2. Proof of the Main Theorem

Now, we need the following theorem

Ambrose Theorem. Let M and M̃ be m-dimensional complete Riemannian manifolds and I : T_{p}M ↦ Tp̃M̃ a linear isometry. Suppose that M is simply connected and for any once broken geodesic γ : [0, l] ↦ M in M

I_{t}(R(u, v)w) = R̃(I_{t}(u), I_{t}(v))I_{t}(w)

for any u, v, w ∈ T_{γ(t)}M, 0 ⩽ t ⩽ l, where R and R̃ denote the curvature tensors of M and M̃, respectively. For any minimal geodesic γ : [0, l] ↦ M with γ(0) = p, define a geodesic γ̃ by γ̃(t) := γ_{I(γ′(0))}(t) and define a map Φ : M ↦ M̃ by Φ(γ(t)) := γ̃(t). Then Φ is well defined and a C^{∞} Riemannian covering. In particular, if M̃ is also simply connected, then M and M̃ are isometric [7].

In our case, since M is complete Kähler, let g and I denote the corresponding Kähler metric and the corresponding complex structure, respectively. Let ∇ and R be the Levi–Civita connection and the curvature tensor of the metric g, respectively. For each p ∈ M, we know that I|_{T_{p}M} : T_{p}M ↦ T_{p}M is a linear isometry, where I|_{T_{p}M} means the restriction of the complex structure I to the tangent space T_{p}M. We see ∇I = 0. Furthermore [5], R(I, I) = R( , ) and I ∘ R = R ∘ I. For any minimal geodesic γ : [0, l] ↦ M with γ(0) = p, define a map Φ_{p} : M ↦ M by

Φ_{p}(γ(t)) := γ_{I(γ′(0))}(t) for t ∈ [0, l].

Then, by Ambrose Theorem, Φ_{p} is an isometry for each p ∈ M.

Proposition 1. Φ_{p}(N) = N for each p ∈ N.

Proof. Firstly, we claim Φ_{p}(N) ⊃ N. For any q ∈ N, there exists a minimal geodesic curve γ : [0, 1] ↦ N such that γ(0) = p and γ(1) = q. By the hypothesis, γ is also a minimal geodesic curve in M. Since Φ_{p} is isometric and N is complex, Φ_{p}^{k} ∘ γ is minimal in N for each k ∈ {1, 2, 3, 4}. Hence,

q = (Φ_{p}^{4} ∘ γ)(1) = Φ_{p}(Φ_{p}^{3}(γ(1))) ∈ Φ_{p}(N).

Secondly, we claim Φ_{p}(N) ⊂ N. For any q ∈ Φ_{p}(N), by definition, there exists a point q̃ ∈ N such that Φ_{p}(q̃) = q. Choose a minimal geodesic curve γ : [0, 1] ↦ N such that γ(0) = p and γ(1) = q̃. Then, γ is also minimal in M. As the above, Φ_{p} ∘ γ is minimal in N. Thus, (Φ_{p} ∘ γ)(1) = q ∈ N. This completes the proof. □

Proof of the Main Theorem. Since (M, I) is a complex manifold, there exists an atlas {(z_{α}, U_{α}) | α ∈ A} of M, being a subfamily of the maximal atlas of M, such that

(i) {U_{α} | α ∈ A} is a locally finite open covering of M,

(ii) there exists a partition of unity {φ_{α} : M ↦ ℝ | α ∈ A} such that supp φ_{α} ⊂ U_{α} for all α ∈ A.

Let π : T M ↦ M be the natural projection map, given by π(p, v) = p for (p, v) ∈ T M. Conveniently, identify the tangent space T M with the holomorphic tangent space T′M [5]. Given a chart z_{α} : U_{α} ↦ ℂ^{m}, α ∈ A, we can naturally have the corresponding chart dz_{α} : T′U_{α} ↦ ℂ^{m} × ℂ^{m} by

dz_{α}(v) = (z_{α}^{1}, z_{α}^{2}, …, z_{α}^{m}; ξ_{α}^{1}, ξ_{α}^{2}, …, ξ_{α}^{m}), where v = ∑_{k=1}^{m}ξ_{α}^{k} \frac{∂}{∂z_{α}^{k}} ∈ T′_{p}U_{α} with p ∈ U_{α}.

For v′, w′ ∈ T′_{v}(T M) with v ∈ T U_{α}(≡ T′U_{α}) and α ∈ A let their coordinate representations be (v′_{α1}, …, v′_{αm}; η_{α1}, …, η_{αm}) and (w′_{α1}, …, w′_{αm}; η′_{α1}, ⋯, η′_{αm}). Then we put

h(v′, w′) := ∑_{\begin{matrix}α∈A \\ v∈T U_{α} \\ i∈\{1,...,m\} \end{matrix}} φ_{α}(p)(v′_{αi}\overline{w′_{αi}} + η_{αi}\overline{η′_{αi}} ),

where p = π(v). This defines a Hermitian metric on the complex manifold T M. Let G be the Riemannian metric on T M which is naturally induced from the Hermitian metric h.

Assume that ρ is differentiable at v ∈ U_{ν} ∩ T_{p}M. By definition, the differential dΦ_{p} of the map Φ_{p} has the following properties

(dΦ_{p})_{p}(v) = Iv and (dΦ_{p})_{p}(Iv) = I(Iv) = −v.

We know that ρ is differentiable at v ∈ U_{ν} if and only if for any unit speed smooth curve c : (−ϵ, ϵ) ↦ U_{ν} with c(0) = v and ϵ > 0 the following limit exists:

lim_{t→0} \frac{ρ(c(t)) − ρ(c(0))}{t}.

Take any unit speed smooth curve c̃: (−ϵ, ϵ) ↦ U_{ν} with c̃(0) = Iv and sufficiently small ϵ > 0. Let p_{t} := π(−Ic̃(t)) for each t ∈ (−ϵ, ϵ). By Proposition 1,

d(N, γ_{c̃(t)}(s)) = d(Φ_{p_{t}}(N), Φ_{p_{t}}(γ_{−Ic̃(t)}(s))) = d(N, γ_{−Ic̃(t)}(s))

for all s ∈ [0, l_{t}] with l_{t} := sup{r > 0 | γ_{−Ic̃(t)}|_{[0,r]} is minimal} so that

ρ(c̃(t)) = sup{s > 0 | d(N, γ_{−Ic̃(t)}(s)) = s} = ρ(−Ic̃(t))

for t ∈ (−ϵ, ϵ). Note that −Ic̃(t) is a unit speed smooth curve in U_{ν} with the property −Ic̃(0) = v. Thus, by the hypothesis, the following limit

lim_{t→0} \frac{ρ(c̃(t)) − ρ(c̃(0))}{t} = lim_{t→0} \frac{ρ(−Ic̃(t)) − ρ(−Ic̃(0))}{t}

exists. Hence, ρ is differentiable at Iv. Furthermore, from this result, ρ is also differentiable at I(Iv) = −v. Therefore, we complete the proof. □

Remarks. 1. In particular, if ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at w ∈ {v, Iv, I^{2}v = −v, I^{3}v = −Iv}.

2. Let ⟨Φ_{p}⟩ be the group generated by the element Φ_{p}. Then ⟨Φ_{p}⟩ is a cyclic group of order 4. Let G := ⋃_{p∈M}⟨Φ_{p}⟩. Then G ⊂ iso(M), where iso(M) denotes the group of all isometries of M

3. For each p ∈ M, let N = {p} as a 0-dimensional complex submanifold of M. Then U_{ν} = U_{p}M, where U_{p}M denotes the unit tangent vector space of M at p. If ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at w ∈ {v, Iv, −v, −Iv}.

4. Consider the complex projective space ℙ^{n} with the Fubini–Study metric [3]. Let ℙ^{k} := {(z_{0} : ⋯: z_{k} : 0 : ⋯: 0) | z_{i} ∈ ℂ, 0 ⩽ i ⩽ k} ⊂ ℙ^{n} for k = 1, …, n − 1. Then ℙ^{n} is a simply connected complete Kähler amnifold and ℙ^{k} is a closed complete totally geodesic complex submanifold of ℙ^{n} such that every minimal geodesic in ℙ^{k} is minimal in ℙ^{n} [4]. Let U_{ν} be the unit normal bundle of ℙ^{k} in ℙ^{n}. If ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at w ∈ {v, Iv, −v, −Iv}.

5. Let (M, g) be a simply connected complete Riemannian manifold with a hyperkähler structure (g, I, J, K) and N a closed complete totally geodesic trianalytic submanifold of M such that every minimal geodesic in N is minimal in M ([3], [9]). If ρ is differentiable at v ∈ U_{ν}, then ρ is also differentiable at w ∈ {R^{i}v | i ∈ {1, 2, 3, 4}, R ∈ S^{2}}, where S^{2} := {aI + bJ + cK | a^{2} + b^{2} + c^{2} = 1}.

Now, we consider

Question 1. Let M be a simply connected complete Kähler manifold and N a closed complete totally geodesic complex submanifold of M. Then, is it true that every minimal geodesic in N is also minimal in M?

The author believes that it may be true, but can not prove it.

Question 2. Let (M, g, I) be a 2-dimensional simply connected complete Kähler manifold and N a 1-dimensional closed complex submanifold of M. Let U_{ν} be the unit normal bundle of N in M. Then, at which v ∈ U_{ν} is ρ : U_{ν} ↦ ℝ differentiable?

Note that if v ∈ T_{p}N with g(v, v) = 1 and u ∈ T_{p}M ∩ U_{ν} for p ∈ N, then we easily get

T_{p}M = ℝ⟨v, Iv, u, Iu⟩ and T_{p}M ∩ U_{ν} = {au + bIu | a^{2} + b^{2} = 1}.

References

[1] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin–Heidelberg–New York, 1969.

[2] J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc. 353 (2001), 21–40.

[3] D. D. Joyce, Compact Manifolds With Special Holonomy, Oxford University Press, 2000.

[4] Y. H. Kim and S. Maeda, A practical criterion for some submanifolds to be totally geodesic, Monatsh. Math. 149 (2006), 233–242.

[5] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, John Wiley and Sons, 1969.

[6] F. Morgan, Geometric Measure Theory, A Beginner’s Guide, Academic Press, 1988.

[7] T. Sakai, Riemannian Geometry, Translation of Mathematical Monographs, vol. 149, Amer. Math. Soc., 1996.

[8] M. Tanaka, Characterization of a differentiable point of the distance function to the cut locus, J. Math. Soc. Japan 55 (2003), 231–241.

[9] M. Verbitsky and D. Kaledin, Hyperkähler Manifolds, Internatioanal Press, 1999.

[10] J. H. C. Whitehead, On the covering of a complete space by the geodesics through a point, Ann. of Math. 36 (1935), 679–704.

School of Mathematical Sciences (Received 27 03 2007) Seoul National University (Revised 15 01 2008) Seoul 151-747 Korea

parkksn@math.snu.ac.kr