ON RICCI CURVATURE AND GEODESICS

B EUGENIO CALABI

1. Introduction. The relationship in a Riemannian manifold between thenumerical invariants of curvature at each point and the global metric propertiesof the manifold has been for a long time a subject for study by many people.In 1941 S. B. Myers [4], extending an earlier result by J. L. Synge, proved that,if the Ricci (mean) curvature of a manifold is uniformly bounded from below bya positive constant c2, then every geodesic arc of length >_ r/c contains conjugatepoints; in particular, if the manifold is complete, its diameter is bounded byv/c. A qualitative generalization of this result was published by W. Ambrose[1] in 1957, his statement reading as follows: Let M be a complete, n-dimensionalRiemannian manifold (n >_ 2); suppose that there exists a point po lYl withthe property that, for every infinite geodesic ray F issued from po and parametri-zed by its arc length s, the following improper integral diverges,

lim inf K(s) ds - oo,

where K(s) is the Ricci curvature at the tangent vector of r at the point cor-responding to arc length s; then each r contains (infinitely many) conjugatepoints of po, and consequently M is compact.The present article contains a different sort of extension of Myers’ theorem;

the following statement arises as a consequence of the main result, which isstated as Theorem 2 in the next paragraph.

THEOREM 1. Let NI be a complete, n-dimensional Riemannian mani/old withnonnegative Ricci curvature everywhere. I], ]or some point poe M, every geodesicray F issuing ]rom Po has the property that

(1.1) lim sup //K(s) ds 1/2 log a

then M is compact; K(s) denotes the Ricci mean curvature of M at the tangentvector o/ F over the point at arc distance s ]rom po

The above theorem and Ambrose’s earlier one contain the same conclusionunder two apparently similar assumptions; neither of these two sets of assump-tions contains the other, so that these two theorems can be used iointly asdependently sufficient conditions for compactness. The integral (1.1) overgeodesic arcs without coniugate points, however, is of independent interest;a sharp estimate of such an integral has been used implicitly before [3, proof of

Received October 31, 1966. This paper was prepared while the author was supported inpart by a grant from the National Science Foundation under Contract GP-4503.

667

668 EUGENIO CALABI

Theorem 2] and can be applied again in at least two other instances known tothe author. It is useful therefore to present here separately the results pertain-ing to estimates of that integral. Other geometric properties of Riemannianmanifolds with nonnegative Ricci curvature can be found in [2], [5].

2. Geodesics without conjugate points. Throughout this paper lYl will denotea Riemannian manifold; completeness is not assumed. We use, in general thetensor notation of Eisenhart, with occasional abbreviations by means of moresynthetic notations.

Let R denote the components of the Riemann curvature tensor; the Riccitensor has components R R, which we define with opposite sign fromEisenhart, by means of the contraction

Ri R Rihi iih

If (0/0x) is a nonzero tangent vector at a point of M, we define the Ricci,or mean, curvature of lYI at , to be

--11 (R,( f; )((f;)O()

If x o’(s) describes locally, in terms of coordinates (x) a differentiable pathII in M parametrized by it arc length s, a

_s

_b, we let

(dx 0) 1 dx’ dx(2.1) g(s) Q\ ds x= R,

n- 1 ds ds

denote the Ricci curvature of M at the tangent vector of II corresponding to thevalue s of the parameter. For the present purposes, the inclusion of an indica-tion of the dependence of K(s) on II seems superfluous. We state now the mainresult of this paper.

THEOREM 2. Let lYI be an n-dimensional Riemannian mani]old with non-negative Ricci curvature everywhere, and let F be an open geodesic arc in M, param-etrized by its arc length s, the latter ranging over the open interval (A, B) where

_A < B

_. Suppose that F contains no pair o] conjugate points; then

]or every compact subinterval [a, b] o] (A, B) with A < a < b < B

(2.2) fb %ilK(s) ds

<_ 1 --t-1/2 log (B b)(a A)/(B a)(b-- A)

lOg(B_ b)(a-- A)

The cross ratio whose logarithm occurs in (2.2) is to be read as its limiting value,i] either A or B or both are infinite.We recall that a closed segment [po, pl] of a geodesic arc F is said to contain

no pair of conjugate points, if and only if the local variation of that segment

ON RICCI CURVATURE AND GEODESICS 669

of F in the space of arcs ioining Po and pl has strictly positive second variation.Equivalently, this means that the exponential map exppo with center Po anddefined in a neighborhood of the segment [0, ql] C R that maps onto the segment[Po, Pl] C F, is nonsingular everywhere on that segment.

Before proving the two theorems stated above, a few preliminary conceptswill be presented.

DEFINITION 1. Let H (A, B) M be a twice differentiable path, i.e. animmersion of an open segment (A, B) (- A < B

_) into an n-dimen-

sional Riemannian manifold M. A tubular neighborhood ?IH of II is a compositeobject (M, II’, r), where lYU is an n-dimensional Riemannian manifold, H’(A, B) M’ a differentiable path in M’, and r M’ -- M a locally isometric mapof M’ into M, satisfying the following conditions.

1) The immersion II’ is a differentiable imbedding, and is proper, i.e., ]or everycompact F C M, H-I(F) is a compact subset of (A, B);

2) For each (A, B), II(t) o II’(t);3) For each point p M IY(A, B), there exists exactly one geodesic segment

joining p with the trace o]IY, intersecting it at right angles; it also minimizes thegeodesic distance between p and the nearest point in H’(A, B). When no mis-understanding is anticipated, we may use the symbol lYI as a synecdoche denotingthe tubular neighborhood (M’, II’, ).

The proof of existence of tubular neighborhoods of a path is an easy exercise.The advantage of this concept is that statements such as that of Theorem 2about an open differentiable path II (e.g. a geodesic) are equivalent to thecorresponding statement about the lift IY of H to a tubular neighborhood.Working the with II’ instead of II, one avoids the topological difficulties thatmay arise from the fact that II need not be either an imbedding or a propermap. An immediate application of this advantage is seen in the followingstatement.

PROPOSITION 1. Let F be an open geodesic segment in a Riemannian manifoldM, and let r be parametrized by its arc length s (--

_A < s < B

_).

Then a necessary and sucient condition for r to contain no conjugate points isthe following. There exists a tubular neighborhood (M, r’, 7) of F with a func-tion lYI’ ----> R that is differentiable everywhere in lYI’ and satisfying

1) o F’(s) s;(A < s < B)2) The gradient of has absolute value 1, i.e.

(2.3) everywhere in

Proo]. We prove the necessity first. Suppose that F contains no pair ofconiugate points and assume without loss of generality that 0 (A, B). Let(lYI", F’, 7) be a tubular neighborhood of r. For any a (A, B) consider theexponential map expa :R" -- lYI" with center l’(a); this map, by assumption,is defined and nonsingular in a neighborhood Ua of the open segment F" C R"

670 uo

that maps onto F’(A, B) under this exponential map. We use a Cartesiancoordinate system (y, yn) in R so that, exp (yl, 0, 0) r’(a + y,)for B a < y < A a; let Va C Ua be a neighborhood of F’ that is mappeddiffeomorphically under exp onto a tubular neighborhood M’ C M’ of r’(A, B)and define, for a 0, a function M - R by means of the equation

o(q’) a (sgn a). (y,) (exp:’ (q’))

This map is differentiable everywhere except at F’(a), and otherwise satisfies(2.3) everywhere. In particular there is a hypersurface 2 C M’ correspondingto the zero set of a, intersecting r’(A, B) at r’(0) at right angles; the vectorfield, restricted to 2 defines the unit normal vector, oriented so that atr’(0) it points in the direction of F’(0). The second fundamental form ofZo, in terms of this orientation of the normal vector, is given by

O -.We observe that, at the point r’(0) 9o 0 and a dr,(o (d/ds) thus @and @ are both independent of a; furthermore, for any fixed q’ M, we havealso q’ M for b sufficiently close to a and, because of the triangle inequality,

0(R)o(q’) > 0.Oa

Thus, if a > 0 and a increases toward B, the quadratic form O is decreasingat r(0), while if a < 0 and a decreases toward A, O is monotone increasing.On the other hand, again because of the fact that the second variation of the arclength of F is positive between any two of its points, for any a, b satisfying A <a < 0 < b < B, wehave

Oo(r’(o)) < O(r’(0)).

Hence there exists quadratic form 0 on the tngent subspce of M’ t F(O)orthogonl to r such that, for all numbers a, b satisfying A < a 0 b < B,

(2.3) Oa(F’(0)) < 0 < Ob(F’(0)).

Now let 2 be any differentiable, oriented hypersurface imbedded in lYI", in-tersecting r’(A, B) orthogonally and only in r’(0), and with second fundamentalform0 at r’ (0). Consider he productR X 2; and identify R with R X r’ (0)R X 2;, 2; with {0} X 2, and the "wedge" R V 2 with the union R X F’(0) kJ{0} X 2;. Then there exists a differentiable map of a neighborhood W1 of(A, B) V 2; C R X 2 into M" such that

i) is an extension of the imbedding of 2 in lYI’t

ii) for eachq 2;, (R X {q}) W (Aq,Bq) {q} and the map F"(A B) - M" defined by F’(s) 1(s, q) is the geodesic arc, parametrizedby its arc length s (positively oriented) and orthogoaal to 2] at q. Ia particularrr,o)

ON RICCI CURVATURE AND GEODESICS 671

Because of (2.4), the map 1 is regular everywhere in r’(A, B); thereforethere is a smaller neighborhood W C W of (A, B) in R X 2 such that therestriction of to W is a differentiable imbedding of W onto an open tubularneighborhood M’ C M" of r’(A, B); for q k(W), let (q) R be defined tobe the number such that

k-l(q) (t, q’), q’

then this function , defined in the tubular neighborhood (M’, r’, ) C(M", r’, ) of r(A, B), is function with the properties 1) and 2) required inthe statement of the proposition.

Conversely, sssume that a function with these two properties exists in atubular neighborhood (M’, r’, ) of r. Then, for any piecewise differentiablepath II in M’ joining r’(a) and r’(b) with A < a < b < B, we hve, becauseof (2.3),

F’(a) b a.

Equality between the first two members can hold, if and only if II is an integralcurve of he gradient field of ; on the other hand, if the trace of H does notcoincide with F’[a, b], then there exists subinterval of H of length at least 6for some 6 > 0, such that the angle between the trace of II and the normalvector of the hypersurfaces constant satisfies Isin I >- e for some e > 0.From this it easily follows that

ds> b-- a+ (1 icosa[) > b- a-t- /e.1

This fact shows that the second vriation of the length of any bounded, closedsegment of r is positive definite. Thus the proof of the proposition is complete.

DEFINITION 2. Given a Riemannian manifold M and a geodesic arc r(A, B) M parametrized by its arc length s, a geodesic field extension o] r consistso] a tubular neighborhood (M’, r’, ) of r and a smooth tangent vector field 0 inM’ satis]ying:

1) For all s (A, B), 0(F’(s)) dr’(d/ds);2) At each point q’ , M’ iO(q’)[ 1;3) The vector field is irrotational; that is to say, i] locally 0 O(O/Ox), i]

the metric is ds g dxx dx and if O g,Ox, then

O0Ox Ox"

Since the domain M’ is, by Definition 1, contractible and hence simply con-nected, a geodesic field extension is, globally in M’, the gradient field of a func-tion that satisfies the conditions of Proposition 1 in order that r be withoutconjugate points.

672 EUGENIO CALABI

3. Proof of the theorems. We begin by proving Theorem 2. Let (M’, r’, r)be a tubular neighborhood of r admitting a function q) M -- R whose gradientV(I) is a geodesic vector field extension of r’. Without loss of generality, wereplace M and r by M’ and r and then drop the "prime" superscripts.

Let Za for any a e (A, B) denote the hypersurfaee in M defined by the equa-tion q a, The vector field 0 V is, at each q M, the unit normal vectorof the hypersurfaee 2:((R)q)The first fundamental form induced on each Na corresponds to the restric-

tion of the metric to the tangent space of 2: the second fundamental formis represented by

O -VV --.. dx dx

in terms of local coordinates; we associate o it the Weingarten operator on thetangent spaee of each N. with tensor components

B =--.g

the eigenvalues of (B) are the principM curvatures of N Consider now thevariation of he Weingarten operator along an integral curve of grad , i.e.along the orthogonal rajectories of N as a varies. This is nohing else bu theLie derivative of he Censor (B) with respect o 0 VO. Using the Ricciidentities, we obtain

OB +[o, O,Bi O,B

Furthermore, by tking first nd second covrint derivatives on both termsof (2.3), we hve successively

Substituting this ia the previous equation, we get

[e, B]

e"eBB +This ]st equation is the Riccti mtrix equation ssocited with the clssicMform of the Jcobi equation of F. Let us denote by , ,_ the principalcurvatures of ech Z., nd byH H= the first nd second mean curvatures,i.e.

"- 1 1 1 BBH H, B, Hn-- 1 = n-- 1 n-- 1 .= n-- 1

By means of contraction of the Ricci equation bove, we obtMn the equation

10 0H() nt- ---Z--Rh

ON RICCI CURVATURE AND GEODESICS 673

Because of the Cauchy-Schwarz inequality H() _> H, we conclude that themean curvature H stisfies the differential ineqution

0H Hx + Q(O) (Q(O) Ricci curvature t 0).

Restricting the bove inequality to F, we set H(s) H(F(s)) for A s B;we have thus

(3.1) dill > (H) + K(s)

where K(s) is defined in (2.1).By comparison with the inequution (3.1) we deduce immediately the fol-

lowing statement, that we formalize s lemma.

LEMMA. Let F (A, B) M be a geodesic without conjugate points in aRiemannian manifold M, parametrized by its arc length s, A < s < Band denote by K(s) the Ricci mean curvature of M at the tangent vector of F overF(s). Then there exists a differentiable function h(s), globally defined over the openinterval (A, B) that satisfies the Riccati equation

(3.2) d h + K(s)ds

From the bove lemma one obtuins immediately Myers’ theorem [4] underthe assumption K(s) c > 0; it is also possible, but not so easy, to deduceAmbrose’s result. In proving our Theorem 2, we shall limit ourselves to theassumption that there is a function h(s) in the open interval (A, B) stisfyingthe inequality

dh > h.(3.3) ds

the conclusion consists of n estimate of an upper bound for the integral

for any compact subinterval [a, b] of (A, B), in terms of A, a, b, and B. Weconsider the four possible cases that can arise as A and B can be, independently,finite or infinite.

Case 1. A andB . Itisclearthttherecanbenofunctionh(s) on the whole line satisfying (3.3) other than h(s) O. Therefore in thiscase K(s) must be identically zero.

Case 2. A > - and B . We make the following substitutions.Let s A e nd define w(t) by means of

h(s) e-(w(t) 1/2).

674 EUGENIO CALABI

Then the interval (A, B) becomes, in terms of t, the whole real line; inequality(3.3) becomes

dw > w(3.5) d--- -z (- < < )

and the integral (3.4) becomes

(3.6) %/h(s) h2(s) ds %/w’(t) w(O "t" - dt

where the bounded interval [a’, b’] corresponds to [a, b] in terms of the substitu-tion of for s. It follows from (3.5) that w(t) is uniformly bounded for all by

-1/2 <_ (t) _< 1/2.

Applying the Schwarz inequality to (3.6), we have

<_ (b’ a’) (w’(t) w(t) -f- -) dt

<_ (b’ a’)(w(b’) w(a’) + -(b’ a’)) (1/2(b’ a’) + 1) I.

Thus we obtain, by rewriting in terms of s and h(s),

f.b %l-if(s) h2(s) ds

<_ %/(1/2(b’ a’) -f- 1)

Case 3. A - andB < .1 1 -J-1/21og-_ 1

We replace s, A, B and h(s) respectivelyby -s, fi -B, / o and h(s) by f(r) -h(-s), reducing this toCase 2.

Case 4. -, <: A B . We again make a substitution of the finiteinterval by the whole real line as follows" let L B A,

A+B Ls

2 A- tanh t, and define w(t) by means of

2h(s) - (w(t) -4- tanh t) cosh t.

Then inequality (3.3) is equivalent to

dw(t) > w.(t)_ 1 (- < < )(3.7) dt

and (3.4) becomes, after appropriate substitution of the limits [a’, b’] for [a, b]

ON RICCI CURVATURE AND GEODESICS 675

(3.8) %/-(s) h2(s) ds /W-(t) w2(t) + 1 dt.

It follows from (3.7) that, in this case, w(t) is uniformly bounded by -1

_w(t)

_1. Applying then the Schwarz inequality to (3.8) we get

(0--(0 + ld N (b’- #) (’(0 + 1-’(O) d

(b’ a’)(w(b’) w(a’) + b’ a’)

(b’--a’)(1 + 1 + b’-a’)

( + b’- a’) .Thus we have

%/-(s) h2(s)ds

_((1 + b’ a’) 1)

"-((1-[- arctanh2b A arctanh2a-- A B) 1}t

{(l -t- 1/21og (B a)(b A))" }’.(B-- b)(a- A) 1

This concludes the proof of Theorem 2.We now show that Theorem i is n esy consequence of Theorem 2. Suppose

that lYI is complete, has positive semidefinite Ricci curvature everywhere, andis not compact. Then for every point po M there exists a geodesic ray issuingfrom po, that achieves the shortest distance between any two of its points, andextends to infinity: in particular this ray has no conjugate points. Let itbe parametrized by s, where 0 < s < . By virtue of Theorem 2, for eachpair of real numbers e, a with 0 < e < a < ,

f," %/K(s)< + { log _ads 1

which clearly contradicts (1.1).We shall construct an example that shows the extent to which the estimate

for the upper bound of (2.2) is sharp. Consider the standard unit euclidean(n 1)-sphere S"-1 with the usual Riemannian metric form denoted by da2,and let M (0, L) S"-1 be described as the set of pairs (t, ) where 0 << L _< o and o S"-. On M we define the Riemannian metric ds dt +

t(1 t/L) da (for L oo, we replace this last expression by its limit value);this metric has everywhere strictly positive curvature. The coordinate func-tion t, globally defined on M, is a function whose gradient has length equal tounity everywhere; consequently the orthogonM trajectories of the hypersurfaces{t constant} are all geodesics without conjugate points, and the value in-

676 EUGENIO CALABI

crements of the parameter along these geodesics correspond to the arc lengthof each segment. It is also easy to verify that along any one of these geodesics,using as its parameter, we have

LK(t) 4t(L t).,

so that, forO < a < b < o,

f b(L- a)%/K(t) dt 1/2 log a(L b)

which comes within one unit ofand is asymptotically equal to--the estimate(2.2).Other applications of Theorem 2 are contemplated in more specialized situa-

tions as mentioned in the introduction. In these the value 1/2 of the coefficientpreceding the logarithm plays a crucial role.

REFERENCES

1. W. AMBROSE, A theorem of Myers, this Journal, vol. 24(1957), pp. 345-348.2. E. CALABI, An extension of E. Hopf’s maximum principle with an application to Riemannian

geometry, this Journal, vol. 25(1958), pp. 45-56 (Erratum, vol. 26(1959), p. 707.)3. E. CALABI, Improper affne hyperspheres of convex type and a generalization of a theorem by

K. JSrgens, Mich. Math. J., vol. 5(1958), pp. 105-126.4. S. B. MYERS, Riemannian manifolds with positive mean curvature, this Journal, vol. 8(1941),

pp. 401-404.5. K. YANO AND S. BOCHNER, Curvature and Betti Numbers, Ann. of Math. Studies No. 32,

Princeton, 1953.

UNIVERSITY OF PENNSYLVANIA