On conjugate points and geodesic loops in a complete Riemannian manifold

0.14 MB·

by  Shicheng Xu

ON CONJUGATE POINTS AND GEODESIC LOOPS IN A
COMPLETE RIEMANNIAN MANIFOLD
SHICHENGXU
Abstract. Awell-knownLemmainRiemanniangeometrybyKlingen-
4 berg says that if x0 is a minimum point of the distance function d(p,·)
1 to p in the cut locus Cp of p, then either there is a minimal geodesic
0 from ptox0 along whichtheyareconjugate, orthereisageodesic loop
2 at p that smoothly goes through x0. In this paper, we prove that: for
n anypointqandanylocalminimumpointx0ofFq(·)=d(p,·)+d(q,·)in
a Cp, either x0 is conjugate to p along each minimal geodesic connecting
J them, or there is a geodesic from p to q passing through x0. In partic-
3 ular, for any local minimum point x0 of d(p,·) in Cp, either p and x0
2 are conjugate along every minimal geodesic from p to x0, or there is a
geodesic loop atpthatsmoothly goesthroughx0. Earlier resultsbased
]
on injective radius estimate would hold underweaker conditions.
G
D
.
h
t
a 1. Introduction
m
Let M be a complete Riemannian manifold. For a point p ∈ M, let T M
[ p
be the tangent space at p and exp : T M → M be the exponential map.
p p
2
For any unit vector v ∈ T M, let σ(v) be the supremum of l such that the
v p
9 geodesic exptv : [0,l] → M is minimizing, κ(v) be the supremum of s such
4 that there is no conjugate point of p along exp tv : [0,s) → M. Let
p
5
5 C˜ = {σ(v)v : for all unit vector v ∈T M}
p p
.
1
0 be the tangential cut locus of p,
4
1 J˜p = {κ(v)v : for all unit vector v ∈ TpM}
:
v be the tangential conjugate locus, and C = expC˜ ⊂ M be the cut locus of
i p p
X p. Let
r D˜ = {tv|0 ≤ t < σ(v), for all unit vector v ∈T M}
a p p
be the maximal open domain of the origin in T M such that the restriction
p
exp | of exp on D˜ is injective. Any geodesic throughout the paper is
p D˜p p p
assumed to be parametrized by arclength. A geodesic γ : [0,l] → M is
called a geodesic loop at p if γ(0) = γ(l) = p. Let d(·,·) be the Riemannian
distance function on M. In [8] Klingenberg proved a lemma that is well-
known now in Riemannian geometry and particularly useful in injectivity
radius estimate.
Date: January 24, 2014.
Keywords: geodesic, cut point,conjugate point, injectivity radius.
2000 MSC: Primary 53C22; Secondary 53C20.
Project 11171143 supported byNational Natural Science Foundation of China.
1
CONJUGATE POINT AND GEODESIC LOOP 2
Lemma 1.1 (Klingenberg [8, 9]). If x ∈ C satisfies that d(p,x ) =
0 p 0
d(p,C ), then either there is a minimal geodesic from p to x along which
p 0
they are conjugate, or there exist exactly two minimal geodesics from p to
x that form a geodesic loop at p smoothly passing through x .
0 0
Recently this lemma was generalized to the case of two points ([7]). Let
p,q be two points in a complete Riemannian manifold M such that C 6= ∅
p
and q 6∈ C . Let F : C → R be a function defined on C by F (x) =
p p;q p p p;q
d(p,x)+d(x,q). Innami, Shiohama and Soga proved in [7] that
Lemma 1.2 ([7]). If J˜ ∩ C˜ = ∅, then for any minimum point x ∈ C
p p 0 p
of F (x) = d(p,x)+d(x,q), there exist a geodesic (and at most two) α :
p;q
[0,F (x )] → M from p to q such that α(d(p,x )) = x .
p;q 0 0 0
Notethatifp = q,thenLemma1.2isreducedtothecaseofKlingenberg’s
Lemma. Inthis paper,weimprove bothresultsin theabove tothefollowing
theorems whose constraints are sharp in general.
Theorem A (Generalized Klingenberg’s Lemma). Let M be a complete
Riemannian manifold and p,q ∈ M such that C 6= ∅ and q 6∈ C . Let x ∈
p p 0
C such that F (x ) = d(p,x )+d(q,x ) is local minimum of F in C .
p p;q 0 0 0 p;q p
Then either p and x are conjugate along every minimal geodesic connecting
0
them, or there is a geodesic (and at most two) α :[0,F (x )]→ M from p
p;q 0
to q such that α(d(p,x )) = x .
0 0
Theorem B (Improved Klingenberg’s Lemma). Let M be a Riemannian
manifold and p be a point in M. Let x ∈ C such that d(p,x ) is a local
0 p 0
minimum of d(p,·) in C . If there is a minimal geodesic from p to x along
p 0
which p is not conjugate to x , then there are exactly two minimal geodesics
0
from p to x that form a whole geodesic smoothly passing through x .
0 0
Moreover, if d(p,x ) is also a local minimum of d(x ,·) in C , then the
0 0 x0
two minimal geodesics form a closed geodesic.
Motivated by Theorem B, we will call a cut point q of p an essential
conjugate point of p if p is conjugate to q along every minimal geodesic con-
necting them. Geodesic loops may not exist if there are essential conjugate
points. For example, a complete noncompact Riemannian manifold of pos-
itive sectional curvature always contains a simple point p (i.e. there is no
geodesic loop at p) whose cut locus is nonempty [6]. By Theorem B, every
local minimum point of d(p,·) in C must be essentially conjugate to p.
p
The conjugate radius had been involved in the injectivity radius estimate
besides the length of geodesic loops (see [8, Lemma 4] or [1, Lemma 1.8]).
Recall that the conjugate radius at p is defined by
conj(p) = min{κ(v)| for all unit vector v ∈ T M}
p
and the conjugate radius of M, conj(M) = inf conj(p). The injectivity
p M
∈
radius of p is defined by
injrad(p) = min{σ(v)| for all unit vector v ∈ T M}
p
and injectivity radius of M, injrad(M) = inf injrad(p). It follows from
p M
∈
Theorem B that the conjugate radius in the injectivity radius estimate can
be replaced by the distance from p to its essential conjugate points. Let
CONJUGATE POINT AND GEODESIC LOOP 3
Jε = C \exp (C˜ \J˜)) be the the set consisting of all essential conjugate
p p p p p
points of p. Let
d(p,Jε) if Jε 6= ∅,
conjε(p) = p p
(+∞ otherwise.
and conjε(M) = min conjε(p). Then the injectivity radius can be ex-
p M
∈
pressed in the following way, where non-essential conjugate points are cov-
ered by geodesic loops.
Theorem 1.3.
conjε(p),
injrad(p)= min ;
half length of the shortest geodesic loop at p
(cid:26) (cid:27)
conjε(M),
injrad(M) = min .
half length of the shortest closed geodesic in M
(cid:26) (cid:27)
By Theorem 1.3, in this paper we call conjε(p) the essential conjugate
radius of p and conjε(M) the essential conjugate radius of M.
There have been rich results in Riemannian geometry where an upper
sectional curvature bound K shows up to offer a lower bound π of the
√K
conjugate radius, which now could be weakened to the essential conjugate
radius. Forinstance,oneisabletogeneralizeCheeger’sfinitenesstheoremof
diffeomorphism classes to Riemannian manifolds whose essential conjugate
radius has a lower bound but sectional curvature has no upper bound.
Theorem 1.4. For any positive integer n, real numbers D,v,r > 0 and
k, there are only finite C -diffeomorphism classes in the set consisting of
∞
n-dimensional Riemannian manifolds whose sectional curvature is bounded
below by k, diameter ≤ D, volume ≥ v and essential conjugate radius ≥ r.
In general conjε(p) 6= conj(p). RPn with the canonical metric is a trivial
example. An immediate question is, when conjε(p) = conj(p)? Next two
applications of Theorem B offers examples where conjε(p) coincides with
conj(p).
Theorem 1.5. Let a point p be the soul of a complete noncompact Rie-
mannian manifold of nonnegative sectional curvature in sense of [4]. Then
either the nearest point to p in C is an essential conjugate point, or exp :
p p
T M → M is a diffeomorphism.
p
By Theorem B, Theorem 1.5 directly follows from the fact that the soul
is totally convex ([4]). For a closed Riemannian manifold M, the radius of
M is defined by rad(M) = min max{d(p,x)| for any x ∈ M}.
p M
∈
Theorem 1.6. Let M be a complete Riemannian manifold whose sectional
curvature ≥ 1 and radius rad(M) > π. Then either there are two points
2
p,q ∈ M of distance d(p,q) = d(p,C ) = d(q,C ) < rad(M) and p,q are
p q
essentially conjugate to each other, or M is isometric to a sphere of constant
curvature π2 .
rad2(M)
Xiaproved([11])thatifthemanifoldM inTheorem1.6satisfiesconj(M) ≥
rad(M) > π, then M is isometric to a sphere of constant curvature. Theo-
2
rem1.6follows hisproofafter replacingKlingenberg’slemmatoTheoremB.
CONJUGATE POINT AND GEODESIC LOOP 4
In particular, if rad(p) = max d(p,x) > π, then injrad(p) = conjε(p) =
x M 2
∈
conj(p). Weaker versions of Theorem 1.6 can also be found in [10].
Theorem A provides a new characterization on general Riemannian man-
ifolds, which is an improvement of Theorem 1 in [7].
Theorem 1.7. Let M be a complete Riemannian manifold and p be a point
in M such that C 6= ∅. Then
p
(1.7.1) either p has an essential conjugate point;
(1.7.2) or there exist at least two geodesics connecting p and every point
q ∈ M (regarding the single point p as a geodesic when p = q).
We now explain the idea of our proof of Theorem A. Let p,q ∈ M such
that C 6= ∅, q 6∈ C and x ∈ C is a minimum point of F in C . Let
p p 0 p p;q p
us consider the special case that there is a unique minimal geodesic [qx ]
0
connecting q and x . A key observation from [7] is that the level set of
0
{F ≤ C} is star-shaped at both p and q. In particular, if two minimal
p;q
geodesics [px ] and [x q] from p to x and from x to q are broken at x ,
0 0 0 0 0
then for any point x 6= x in [px ] and any minimal geodesic [qx] connecting
0 0
q and x, we have [qx]∩ C = ∅. Thus [qx] admit a unique lifting [qx] in
p
D˜ . If two minimal geodesics, say [px ] and [px ] , do not form a whole
p 0 1 0 2
geodesicwith[x q]atthesametime,thenbymovingxtox along[pxg] and
0 0 0 1
[px ] respectively, one may expect two liftings of [qx ] in D˜ with different
0 2 0 p
endpoints when partial limits of [qx] exist.
Under the condition that J˜ ∩C˜ = ∅ as in Lemma 1.2 and [7], all such
p p
minimal geodesics [px] are clearlygdefinite away from the tangential conju-
gate locus. So we are able to take limit to meet a contradiction, for the lift
]
[qx ] of [qx ] is unique in the closure of D˜ .
0 0 p
Inthe general case onedoes notknow whether[qx ] admits a lifting at its
0
endpoint in the closure of D˜ , nor the liftings [qx] have a partial limit as x
p
approaches x along [px ]. We will prove that, if there is a minimal geodesic
0 0
α from p to x along which they are not conjuggate to each other, and the
0
union of α and [qx ] is broken at x , then [qx ] always has a lifting in the
0 0 0
closure of D˜ , which share a common endpoint with α˜ (see Lemma 2.2).
p
The uniqueness of the minimal geodesic [qx ] is not an essential problem,
0
becauseby(2.2.4)and(2.2.5)wearealways abletomove q along[qx ]while
0
keeping F minimal at x in C . A local minimum of F can be reduced
p;q 0 p p;q
to the minimum case similarly.
The detailed proof of Theorem A will be given in the next section.
2. Proof of the generalized Klingenberg’s lemma
InthissectionwewillproveTheoremA.LetM beacompleteRiemannian
manifold and p,q be two points in M such that C 6= ∅ and q 6∈ C . Let us
p p
consider the function
F :C → R, F (x) = d(x,q)+d(x,p)
p;q p p;q
and assume that F takes its minimum at x ∈ C . Because q is not a
p;q 0 p
cut point of p, q 6= x . Let γ : [0,d(q,x )] → M be a minimal geodesic
0 0
CONJUGATE POINT AND GEODESIC LOOP 5
from q = γ(0) to x = γ(d(q,x )). Because x is a cut point of p, for any
0 0 0
0 ≤ t < d(q,x ) we have
0
F (γ(t)) = d(q,γ(t))+d(γ(t),p)
q
< d(q,γ(t))+d(γ(t),x )+d(x ,p) (2.1)
0 0
= d(q,x )+d(x ,p)
0 0
= F (x )
q 0
= minF .
q
Therefore γ(t) (0 ≤ t < d(q,x )) is not a cut point of p, and we are able to
0
lift γ| to (exp | ) 1◦γ uniquely in the tangential segment domain
[0,d(q,x0)) p D˜p −
D˜ ⊂ T M, where D˜ is the maximal open domain of the origin in T M
p p p p
such that the restriction exp | of exp on D˜ is injective.
p D˜p p p
Wenowprovethat,ifx ∈ exp (C˜(p)\J˜(p)),theneitherγ canbeliftedon
0 p
the whole interval [0,d(q,x )] such that the endpoint of the lift is a regular
0
point of exp , or γ can be extended to a geodesic from q to p that goes
p
through x .
0
Lemma 2.2. Assume that there is a minimal geodesic α :[0,d(p,x )]→ M
0
from p = α(0) to x = α(d(p,x )) along which p is not conjugate to x . Let
0 0 0
w = d(p,x )α(0) ∈ T M, where α(0) is the unit tangent vector of α at p.
0 ′ p ′
Then
(2.2.1) either γ and α form a whole geodesic at x ;
0
(2.2.2) or there is a unique smooth lift γ˜ : [0,d(q,x )] → T M of γ :
0 p
[0,d(q,x )] → M in the tangential segment domain D˜ ⊂ T M such
0 p p
that γ˜(0) = (exp | ) 1γ(0) and γ˜(d(q,x )) = w.
p D˜p − 0
Proof. We first prove the case that γ is the unique minimal geodesic from q
to x . Assumethat γ and αdonot forma wholegeodesic at x . Let {α(s )}
0 0 i
(0 < s < d(p,x )) be a sequence of interior points of α that converges to
i 0
x as i → ∞, and let γ : [0,d(q,α(s ))] → M be a minimal geodesic from
0 si i
q to α(s ). Then γ converges to γ as i → ∞.
i si
Because γ and α are broken at x , by the triangle inequality and similar
0
calculation in (2.1), for each s , we have
i
F (α(s )) < F (x ) = minF ,
p;q i p;q 0 p;q
and thus for any 0≤ t ≤ d(q,α(s )),
i
F (γ (t)) ≤ F (α(s )) < minF , (2.2.3)
p;q si p;q i p;q
which implies that none of points in γ is a cut point of p. Therefore there
si
is a unique lift curve γ˜ of γ starting at (exp | ) 1γ(0) in the tangential
si si p D˜p −
segment domain D˜ ⊂ T M.
p p
If {γ˜ } is uniformly Lipschitz, then by the Arzel`a-Ascoli theorem, a sub-
si
sequence of {γ˜ } converges to a continuous curve γ˜ :[0,d(q,x )]→ T M,
si 0 p
∞
which satisfies that
γ˜ (0) = (exp | ) 1γ(0), γ˜ (d(q,x )) = w,
∞ p D˜p − ∞ 0
and
exp (γ˜ (t)) = lim γ (t) = γ(t), for all 0 ≤ t ≤ d(q,x ).
p ∞ i si 0
→∞
CONJUGATE POINT AND GEODESIC LOOP 6
That is, γ has a unique lifting in the segment domain that satisfies (2.2.2).
Toprove that{γ˜ } is uniformly Lipschitz, it suffices to show that there is
si
N > 0 such that the distance between γ˜ ([0,d(q,α(s ))]) and tangen-
i N si i
tial conjugate locus J˜ ⊂ T M is positive≥. Indeed, because γ˜ (d(q,α(s )))
p p S si i
converges to w = d(p,x )α(0), at which the differential d(exp ) is non-
0 ′ p
singular, there is a small δ > 0 and some ǫ >0 such that
d(γ˜ (t),J˜)> ǫ, for all d(q,α(s ))−δ ≤ t ≤ d(q,α(s )) and large i.
si p i i
On the other hand, because the restriction γ | converges to
si [0,d(q,α(si)) δ]
γ| , which lies in the segment domain D = exp−D˜ ⊂ M, there
[0,d(q,x0) δ] p p p
is some ǫ−> 0 such that
1
d(γ˜ (t),J˜) > ǫ , for all 0 ≤ t ≤ d(q,α(s ))−δ and large i.
si p 1 i
Nowwhatremainsistoprovethecasethattheminimalgeodesicfromq to
x isnotunique. Letusfixsomeinterior pointq = γ(t )(0 < t < d(q,x ))
0 1 1 1 0
of γ and consider the function F :C → R instead. Because
p;q1 p
F (x) = d(q ,x)+d(x,p)
p;q1 1
≥ d(x,q)−d(q,q )+d(x,p)
1
= F (x)−d(q,q ) (2.2.4)
p;q 1
and
F (x ) = d(x ,q)−d(q,q )+d(x ,p)
p;q1 0 0 1 0
= F (x )−d(q,q ), (2.2.5)
p;q 0 1
we see that F (x) also takes minimum at x . Now the minimal geodesic
p;q1 0
from q to x is unique. By the same argument as above, either γ|
1 0 [t1,d(q,x0)]
from a whole geodesic with α at x , or it has a unique lift in the tangential
0
segment domain. So does γ. (cid:3)
A local minimum point of F can be reduced to the case of global mini-
p;q
mum by the following lemma.
Lemma 2.3. Let x be a local minimum point of F : C → R in C ,
0 p;q p p
and γ : [0,d(q,x )] → M be a minimal geodesic from q = γ(0) to x =
0 0
γ(d(q,x )). Then for any interior point q = γ(t) of γ that is sufficient
0 t
close to x , the function F : C → R takes its minimum at x .
0 p;qt p 0
Proof. By (2.2.4) and (2.2.5), x is also a local minimum point of F
0 p;qt
for any t ∈ [0,d(q,x )). Therefore, it suffices to show that F takes its
0 p;qt
minimum near x as q sufficient close to x .
0 t 0
Let us argue by contradiction. Assuming the contrary, one is able to find
a sequence of points {q = γ(t )} that converges to x such that F takes
i i 0 p;qi
its minimum at some point z ∈ C outside an open ball of x ,
i p 0
B (x )= {x ∈ M|d(x,x ) < ǫ}.
ǫ 0 0
By passing to a subsequence, we assume that z → z ∈ C . Then for any
i 0 p
y ∈ C ,
p
d(p,z )+d(z ,q )= F (z )≤ F (y) = d(p,y)+d(y,q ).
i i i p;qi i p;qi i
CONJUGATE POINT AND GEODESIC LOOP 7
Taking limit of the above inequality, we get
F (z )≤ F (y), for any y ∈ C .
p;x0 0 p;x0 p
Let y = x , then
0
d(p,z )+d(z ,x )≤ F (x )= d(p,x ).
0 0 0 p;x0 0 0
Because d(z ,x ) ≥ ǫ, this implies that z is an interior point of a minimal
0 0 0
geodesic from p to x , and it contradicts to the fact that z ∈ C . (cid:3)
0 0 p
We now are ready to prove Theorem A.
Proof of Theorem A. Let x ∈C be a local minimum point of the function
0 p
F :C → R, F (x) = d(x,q)+d(x,p).
p;q p p;q
Let α : [0,d(p,x )] → M be a minimal geodesic from p = α(0) to x =
0 0
α(d(p,x )),alongwhichpisnotconjugatetox . Letγ : [0,d(q,x )] → M be
0 0 0
aminimalgeodesicfromq = γ(0)tox = γ(d(q,x )), andw = d(p,x )α(0).
0 0 0 ′
First let us prove that there is a minimal geodesic from p to x that
0
forms a whole geodesic with γ. According to Lemma 2.3, by moving q to an
interior point in γ that is sufficient close to x and denoted also by q, it can
0
be reduced to the case that x is a minimal point of F and γ is a unique
0 p;q
minimal geodesic connecting q and x .
0
By Lemma 2.2, if α and γ are broken at x , then γ has a unique lift
0
γ˜ : [0,d(q,x )] → T M in the tangential segment domain D˜ ⊂ T M, whose
0 p p p
endpoint satisfies
γ˜(d(q,x )) = w.
0
Because x is not conjugate to p along α, there is another minimal geodesic
0
β : [0,d(p,x )] → M from p to x . We now prove that β must form a whole
0 0
geodesic with γ.
Assuming the contrary, that is, β does not form a whole geodesic with γ
neither. Foranyinteriorpointβ(s)(0 < s < d(p,x ))inβ,letl = d(q,β(s))
0 s
and γ : [0,l ] → M be a minimal geodesic from q to β(s). Then by the
s s
same argument as (2.2.3), for any 0 ≤ t ≤ l we have
s
F (γ (t)) ≤ F (β(s)) < F (x ) = minF ,
p;q s p;q p;q 0 p;q
which implies that γ has a unique lift γ˜ in the tangential segment domain
s s
D˜ ⊂ T M.
p p
We point it out that, because the endpoint γ˜(l ) of γ˜ may approach
s s
J˜ ⊂ T M,onecannotdirectlyconcludethatthefamilyofcurvesγ˜ contains
p p s
any convergent subsequenceas s → d(p,x ). Instead, let us consider a small
0
metric ball B (w) at w = d(p,x )α(0) in T M on which the restriction of
ǫ 0 ′ p
exp
p
exp :B (w) → M
p Bǫ(w) ǫ
is an diffeomorphism onto its (cid:12)image. Then d(p,x0)β′(0) 6∈ Bǫ(w). Because
(cid:12)
γ converges to γ as s → d(p,x ) and the restriction exp of exp on
s 0 p D˜p p
the tangential segment domain D˜p is a diffeomorphism, γ˜s|[0,l(cid:12)(cid:12)s) converges to
γ˜ | pointwisely, that is, for 0 < t < 1,
s [0,d(q,x0))
γ˜ (l ·t)→ γ˜(d(q,x )·t) as s→ d(p,x ).
s s 0 0
CONJUGATE POINT AND GEODESIC LOOP 8
Then for any 0 < t < 1 that is sufficient close to 1, there exists 0< s(t ) <
1 1
d(p,x ) such that for all s(t ) < s < d(p,x ),
0 1 0
γ˜s(lst1) ∈ Bǫ(w), γ˜s(ls)∈ Bǫ(d(p,x0)β′(0)),
4 2
and γs|[lst1,ls] lies in the open neighborhood exppB4ǫ(w) of x0. Because the
lift of γs|[lst1,ls] is unique in D˜p and γ˜s(lst1) ∈ B4ǫ(w), we conclude that
γ˜s|[lst1,ls] lies in B4ǫ(w), which contradicts to the fact that
Bǫ(w)∩Bǫ(d(p,x0)β′(0)) = ∅.
4 2
Secondly, we prove that there are at most two geodesics from q to p
passing through x . Indeed, if there are two distinct minimal geodesics γ ,
0 1
γ connecting q and x , then one of them, say γ , does not form a whole
2 0 1
geodesic with α. By the proof above, any other minimal geodesic except α
connecting p and x from a whole geodesic with γ . By Lemma 2.2, γ form
0 1 2
awholegeodesic withα. Therefore, thereareexactly two minimal geodesics
α and β from p to x , the union of β and γ form a whole geodesic going
0 1
through x , and α and γ form another whole geodesic. In particular, there
0 2
won’t be a third geodesic from q to x .
0
(cid:3)
TheoremBandTheorem 1.7 areimmediate corollaries of TheoremA.We
give a proof of Theorem 1.4 to end the paper.
Proof of Theorem 1.4. Let M(n,k,D,v,r) be the set consisting of all com-
plete n-dimensional Riemannian manifolds whose sectional curvature ≥ k,
diameter ≤ D, volume ≥ v and essential conjugate radius ≥ r. Let M ∈
M(n,k,D,v,r). By Cheeger’s lemma ([2], see Theorem 5.8 in [3]), there
is an universal constant c (D,v,k) > 0 depending only on n,D,v,k such
n
that every smooth closed geodesic on M has length > c (D,v,k). There-
n
fore, by Theorem 1.3 the injectivity radius of M is bounded below by
min{r, 1c (D,v,k)}. The finiteness of diffeomorphism classes in the set
2 n
M(n,k,D,v,r)follows fromthestandardargument(see[5,12]forexample)
on the construction of a diffeomorphism between two Riemannian manifolds
with small Gromov-Hausdorff distance . (cid:3)
References
[1] U.AbreschandW.T.Meyer.Injectivityradiusestimatesandspheretheorems.Com-
parison geometry, 30:1–47, 1997.
[2] J.Cheeger.Finitenesstheoremsforriemannianmanifolds.Am.J.Math.,92(1):61–74,
1970.
[3] J. Cheeger and D.G. Ebin. Comparison theorems in Riemannian geometry, volume
365. AMS Bookstore, 1975.
[4] J. Cheeger and D. Gromoll. On the structure of complete manifolds of nonnegative
curvature.Ann. of Math., 96(3):413–443, 1972.
[5] K. Fukaya. Collapsing riemannian manifolds to ones of lower dimensions. J. Differ-
ential Geom., 25(1):139–156, 1987.
[6] D. Gromoll and W. Meyer. On complete open manifolds of positive curvature. Ann.
of Math., 90(1):75–90, 1969.
[7] N. Innami, K. Shiohama, and T. Soga. The cut loci, conjugate loci and poles in a
complete riemannian manifold. Geom. Funct. Anal., 22(5):1400–1406, 2012.
[8] W. Klingenberg. Contributions to riemannian geometry in the large. Ann. of Math.,
69(3):654–666, 1959.
CONJUGATE POINT AND GEODESIC LOOP 9
[9] W. Klingenberg. Riemannian geometry. Walter de Gruyter,1995.
[10] Q. Wang. On the geometry of positively curved manifolds with large radius. Illinois
J. Math., 48(1):89–96, 01 2004.
[11] Changyu Xia. A round sphere theorem for positive sectional curvature. Compositio
Math., 142(5):1327–1331, 2006.
[12] T. Yamaguchi. Collapsing and pinching under a lower curvature bound. Ann. of
Math., 133(2):317–357, 1991.
School of Mathematics, Capital Normal University, Beijing, China
E-mail address: [email protected]