Download Convexity of spheres in a manifold without conjugate points

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 112, No. 4, November 2002, pp. 595–599.
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Convexity of spheres in a manifold without conjugate points
AKHIL RANJAN and HEMANGI SHAH
Department of Mathematics, Indian Institute of Technology, Powai,
Mumbai 400 076, India
E-mail: [email protected]; [email protected]
MS received 13 February 2002; revised 17 August 2002
Abstract. For a non-compact, complete and simply connected manifold M without
conjugate points, we prove that if the determinant of the second fundamental form of the
geodesic spheres in M is a radial function, then the geodesic spheres are convex. We also
show that if M is two or three dimensional and without conjugate points, then, at every
point there exists a ray with no focal points on it relative to the initial point of the ray. The
proofs use a result from the theory of vector bundles combined with the index lemma.
Keywords. Index lemma; minimal horospheres; vector bundle; Stiefel–Whitney
classes; focal set.
1. Introduction
Let (M d , g) be a complete Riemannian manifold such that there are no pairs of conjugate
points. The well known cases are manifolds of non-positive sectional curvatures. These
examples have the additional property that there are no focal points either. We say that q
is a focal point of p along a geodesic γ joining them, if there is a Jacobi field J along
γ satisfying J 0 (p) = 0 and J (q) = 0, cf. [3]. See §2 for some explanations. It can
be seen easily that, if a complete Riemannian manifold is free of focal points, then the
geodesic spheres are all convex (i.e., have positive definite second fundamental form) and
consequently there cannot be conjugate points either. The converse is however false, as an
example of a surface constructed by Ballmann et al [1] shows. This surface has no conjugate
points but has focal points. The aim of this paper is to investigate conditions so that the
absence of conjugate points implies the absence of focal points or, more weakly, existence
of rays without focal points. It should be pointed out that if a ray γ (t), 0 ≤ t < ∞, is
without focal points of γ (0), it does not follow that a point γ (t0 ), t0 > 0 will be without
focal points along γ (t), t0 ≤ t < ∞. This is in contrast to the corresponding situation for
conjugate points.
The following are the main results of this paper.
Theorem A. Let (M, g) be a non-compact, complete and simply connected manifold
without conjugate points. Suppose that the determinant of the second fundamental form L
of any geodesic sphere in M depends only on the radius of the sphere. Then, there are no
focal points. Equivalently, all the geodesic spheres in M are convex.
It follows that in these spaces Busemann functions are convex. See [7] for definition and
properties of Busemann functions. As a corollary we show that asymptotically harmonic
manifolds whose horospheres are minimal and for which detL is a radial function are flat.
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Theorem B. Let (M, g) be a non-compact, complete and simply connected manifold
without conjugate points. If M is two or three dimensional, then, at each point p ∈ M
there exists a unit tangent vector u such that there is no focal point of p along the ray
γu (t), 0 ≤ t < ∞.
The main tool of this investigation is the following topological result, first proved by
Szabo [10]. We give a slightly different proof here.
Lemma. Let S n ⊂ IRn+1 be the unit sphere. Let V be any non-trivial sub-bundle of the
tangent bundle T S n of the sphere S n . Then there is an antipodal pair {±q} such that
Vq ∩ V−q is non-trivial. Here Vq denotes the fiber at q and is treated as a subspace of the
ambient Euclidean space in a natural way.
2. Proofs of the results
We begin by proving the topological lemma referred to in the Introduction. For more details
we refer to [6].
Proof of the Lemma.
Case 1. rank V = 1.
In this case V is trivial due to simple connectivity of S n for n ≥ 2. (For n = 1, V = T S 1
and there is nothing to prove.) Hence V admits a nowhere vanishing section X of V .
Consider the field X(p) + X(−p) defined on RP n . Let γ be the canonical line bundle
over RP n . Then X(p) + X(−p) is a nowhere vanishing section of the bundle γ ⊥ in case
Vp ∩ V−p = (0) ∀p.
As defined in [6], γ ⊥ is the orthogonal complement of γ in the trivial bundle εn+1 over
RP n . Let a be the generator of H 1 (RP n , Z2 ), the first cohomology group of RP n with
coefficients in Z2 . Let wi (γ ⊥ ) denote the ith Stiefel–Whitney class of γ ⊥ . It is known,
again from [6], that wi (γ ⊥ ) = a i for 0 ≤ i ≤ n. Therefore, wn (γ ⊥ ) = a n 6= 0. But as
X(p) + X(−p) is a nowhere zero section of the bundle γ ⊥ , by using Proposition 4 of [6],
p. 39, wn (γ ⊥ ) = 0.
Thus we conclude that somewhere this cross section must vanish. Hence there exists a
pair of antipodal points {±p} such that X(p)+X(−p) = 0; which implies that Vp = V−p .
Therefore, Vp ∩ V−p 6= (0).
Case 2. Let r = rankV ≥ 2.
Let S n−1 be any equator. Then V |S n−1 is trivial as S n−1 bounds a contractible hemisphere D n . Let X1 , X2 , ..., Xr be linearly independent sections of V along S n−1 . Suppose
Vp ∩ V−p = (0), ∀p ∈ S n−1 . Then Yi (±p) = Xi (p) + Xi (−p) will give r linearly
independent sections of γ ⊥ |RP n−1 . Hence, again invoking Proposition 4 of [6], p. 39,
wn−r+1 (γ ⊥ |RP n−1 ) = 0. But wn−r+1 (γ ⊥ |RP n−1 ) = a n−r+1 6= 0, if r > 1. Hence a contradiction.
2
Therefore, Vp ∩ V−p 6= (0) for some antipode pair {±p}.
Fix p ∈ M and v ∈ Up M. Let N(v) be the geodesic submanifold defined by v, viz.,
N (v) = expp {v ⊥ }. Suppose γv is the geodesic with γv (0) = p, γv0 (0) = v. Then q = γv (r)
is called a focal point of N(v) along γv if there exists a non-zero Jacobi field along γv with
J 0 (0) = 0, J (r) = 0.
Convexity of spheres in a manifold without conjugate points
597
In other words q is a focal point of N (v) if and only if 0 is an eigenvalue of L where L
is the second fundamental form of S(q, r) at p. Thus detL = 0.
In what follows we simply say that q is a focal point of p. This does not cause any
confusion since there is a unique geodesic from p to q. We also note that p may, or may
not be, a focal point of q.
DEFINITION 2.1
The set of all focal points of p is called the focal set of p.
We remark here that this may not be a standard definition. For convenience we also
adopt the following notation. For a fixed p ∈ M and q 6= p, we let Lq denote the second
fundamental form of S(q, d(p, q)) at the point p. If u ∈ Up M is the unique vector pointing
towards q, then Lq is a symmetric linear map of Tp S(q, d(p, q)) = u⊥ . It has been shown
in [8] that Lq is strictly decreasing as q moves along the ray γu away from p. In particular
there are at most (d − 1) focal points of p (counted with multiplicity) along the ray γu .
Lemma 2.1. Let M be a simply connected manifold without conjugate points. If the focal
set around p is compact, then it is empty. Hence the geodesic spheres through p are convex
at p.
Proof. Choose R to be sufficiently large so that the geodesic sphere S(p, R) (centered at
p and of radius R), encloses all the focal points. Let q ∈ S(p, R). Since q lies outside the
focal set of p, detLq 6= 0 for each q ∈ S(p, R). It follows that the number of negative
eigenvalues (counted with multiplicity) of Lq is a positive constant k throughout S(p, R).
It is easy to see that k is precisely the number of focal points (counted with multiplicity) of
p in each direction from p. For v ∈ Up M, let q(v) ∈ S(p, R) be the unique point where
the geodesic γv cuts S(p, R). Consider
Ev− = linear span{w ∈ v ⊥ : w is a negative eigen vector of Lq(v) }.
Therefore, E = ∪E −
v , where the union is taken over Up M, is a sub-bundle of T Up M of
rank k. Hence using Lemma 2.1 we can find a pair of antipodal points, say v and −v, such
−
6= 0. Consider the unique geodesic γv joining q(v) to q(−v) passing
that w ∈ Ev− ∩ E−v
through p. Consider the unique Jacobi field J1 along γv with J1 (q(v)) = 0 and J1 (p) = w.
Also let J2 be the unique Jacobi field along γ−v with J2 (p) = w and J2 (q(−v)) = 0.
Then from the index lemma [3,8] it follows that
I (J1 ∪ J2 ) = (Lq(v) + Lq(−v) )(w), w < 0.
But this contradicts the fact that geodesic joining q(v) and q(−v) is minimizing. Therefore,
the focal set must be empty. Hence Lq > 0 everywhere and all the geodesic spheres
through p are convex at p.
2
Proof of Theorem A. Since detL is a radial function, focal set of p is a finite union of
geodesic spheres centered at p. Hence the proof follows from the above theorem.
2
COROLLARY 2.1
If (M, g) is an asymptotically harmonic manifold with minimal horospheres such that detL
is a radial function, then M is flat.
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Akhil Ranjan and Hemangi Shah
Proof. In this case there are no focal points or equivalently all the geodesic spheres are
convex. Since the horospheres are limits of geodesic spheres, they are also convex. Since by
hypothesis they are also minimal hypersurfaces, it follows that they are all totally geodesic.
Consider a geodesic γ . The family of horospheres orthogonal to it is a family of parallel
hypersurfaces. Now the Riccati equation (see [8]) L0 + L2 + R = 0 along γ shows that
R = 0 along γ . Since γ is arbitrary, M is flat.
2
Remarks
1. If an asymptotically harmonic manifold with minimal horospheres is also assumed to
be Einstein, for example, a harmonic manifold (see [2] for definition of a harmonic
manifold) with minimal horospheres, then even one ray which is free of focal points
(relative to its initial point), forces M to be flat. This is because the trace of R evaluates
the Ricci curvature.
2. Note that, all symmetric spaces satisfy the condition that detL is a radial function. Since
the Lichenrowicz’s conjecture [2,5] is true in compact simply connected case [9] but
false in the non-compact case [4], it is natural to investigate those harmonic manifolds
which have the additional property that detL is a radial function along with trL.
Proof of Theorem B. Let q1 and q2 be collinear with p but on opposite sides. Then it is
easy to show that (see e.g. [8]) Lq1 + Lq2 is positive definite on {±v}⊥ where v ∈ Up M,
and v points towards q1 . Consequently −v points towards q2 . If M is a surface, v ⊥ is
one dimensional and if there is a focal point of p on γv (t) for t > 0, then for each point
γv (t), Lγv (t) will be positive definite for t < 0. Thus the opposite ray is free of focal
points.
If M is three dimensional with p having a focal point in every direction, we consider
two cases:
Case 1. There is a ray with two focal points (counted with multiplicity).
Case 2. Each ray has exactly one simple focal point.
In the first case, let γv have two focal points, then Lγv (t) will be negative definite for
large positive t and consequently, the opposite ray γ−v will be without focal points. This
proves that Case 1 cannot occur.
In the second case, if each ray has exactly one (simple) focal point, then by the continuity
of the function u 7→ t (u), where t (u) is the distance of the first focal point from p along
the ray γu , it follows that the focal set is enclosed by the geodesic sphere S(p, 2R) where
R = sup{t (u) : u ∈ Up M}. This makes the focal set of p compact and non-empty, a
contradiction. Hence Case 2 is impossible too. This means that the hypothesis which gives
rise to these two cases is false and we are through.
2
Acknowledgements
The authors would like to thank G Santhanam and the referee for bringing ref. [10] to their
notice. The second author thanks National Board for Higher Mathematics for its research
fellowship.
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