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Estimates on the Metric Tensor Estimates on the Metric Tensor David Glickenstein February 14, 2000 1. Intro duction. This is a summary of some easy estimates on the metric tensor of a manifold if the sectional curvatures are b ounded. It generally follows [1, section 1.8]. 2. De nitions. We will estimate derivatives of the metric in geo desic p olar co ordinates. Supp ose wehave a Riemannian manifold M; g. We consider geo desic p olar co ordinates, 2 2 i j and in these co ordinates the metric can b e written as g = dr + r g i j d d . We also are going to need to take a geo desic ton M which is parametrized by arclength, 0 and we will say that P = 0, Q = r , and T t= t. 0 In the sequel, we shall talk ab out vector elds V along the curve . When we write V t, it is understo o d to mean r V t, where r is the Riemannian connection on M . T t We shall also restrict ourselves to r such that r is in the ball around P where exp is P a di eomorphism inside the injectivity radius. 3. Relevant Theorems. We will use Jacobi elds and the Index Inequality. We rst should de ne the Index of a vector eld. De nition 3.1. The index at r of a vector eld V t along a curve is Z r 0 0 I V; V = [gV t;V t + g RT t;V tT t;V t] dt r 0 d where T t= is the tangential vector eld to . dt This is, I think, essentially the second variation of arclength. The imp ortant theorem is the following: Theorem 3.2 Index Inequality. See, for instance, [2]. Let t bea geodesic between P = 0 and Q = r such that therearenoconjugate points between P and Q, and let J t be a Jacobi eld along such that J 0 = 0 and V t be another smooth vector eld d , V 0=0, and J r = Vr, along such that both J and V are orthogonal to T t= dt then I J;J I V; V . r r In order to use the Index Inequality,we will need the following two prop ositions. Prop osition 3.3 See [1], 1.46 . Q = exp X is a conjugate point if and only if exp 0 P P is singular at X . 0 Thus if we stay within the injectivity radius on M ,we will not encounter any conjugate p oints. We will also need to know: 1 2 Prop osition 3.4. The injectivity radius for a spaceofconstant curvature b is =b. Finally,we will use the fact that Jacobi elds are the in nitesimal variations of geo desics. In geo desic normal co ordinates at P , the geo desics through p are given by exp tv . Thus p if wehave a Jacobi eld along a geo desic such that J 0 = 0, then we see that in our @ 0 geo desic p olar co ordinates wehave Jacobi elds J t= tkJ 0k where kw k foravector @ tv p w means g w; w. Thus we see that: Prop osition 3.5. For a Jacobi eld J along a geodesic t, t 2 [0;r] with r =Q, @ 2 0 2 such that J 0=0 and J r = j , we have g J r ;Jr = r g kJ 0k . Q @ 4. Lower Bounds for the Comp onents of the Metric Tensor. We shall prove the following: 2 Theorem 4.1. If the sectional curvatureofMisboundedabove by b then we have the fol lowing two estimates, provided r<=b: p @ 1 g b cotbr log @r r sin br g r; br i where = for any i as de ned in the polar representation of the metric. 2 2 Now supp ose the sectional curvature of M is b ounded ab ovebyb ,sogRJ;TT; J b , 2 or g RT; J T; J b . 00 We furthermore observe that if J is a Jacobi eld, then, by de nition, J t=RT; J T , so we get Z r 0 0 00 0 I J;J= [gJ t;J t + g J t;Jt] dt = g J r ;Jr 4- 1 r 0 We are going to compare the metric on M with the metric on a space of constant curvature 2 b .To do this, we need to make a frame fe t; :::; e tg on M by taking an orthonormal basis 1 n 0 fe ; :::; e g of T M such that e = 0 and then parallel translating it along . Notice 1 n 1 0 0 that since is a geo desic and parametrized by arclength, e t= t. We also need to take 1 ~ a geo desic ~ on another manifold M which is the same length as and do the same thing i i to get a frame fe~ t; :::; ~e tg along ~ .We then have the map : c te t 7! c t~e t, or 1 n i i ~ V 7! V . Notice the following: ~ Lemma 4.2. The map :M; g ! M; g~ has the fol lowing properties for vector elds ~ ~ V and W along and V and W along ~ : ~ ~ g V; W =g ~V;W 0 0 ~ ~ gV ;W=g ~V ;W 0 0 0 0 ~ ~ gV ;W =g ~V ;W i i Proof: Use the frames wehave already set up. Then if V = v e and W = w e where i i i i v ;w are real-valued functions, then X i j i i i j ~ ~ g V; W =v w ge;e = v w =v w g~~e ; e~ =g ~V;W i j i j 2 Notice that it is imp ortant that we use frames instead of coordinates. Furthermore, since 0 0 0 0 i i i 0 i ~ the frames are parallel, i.e. r e = 0,, we get V = v e + v r e = v e and V = v e~ so T i i T i i i the last items follow. 2 Now, Z r 0 0 I J;J = [gJ ;J +gRT; J T; J ] dt 0 Z r 0 0 2 g J ;J b gJ;J dt 0 b ~ ~ = I J; J b 2 where I is the index on a manifold of constant curvature b and where wehavechosen a frame fe g along and a corresp onding frame fe~ g along some curve~ of the same length i i in the manifold of constant curvature. In constant curvature, wehave a Jacobi eld whichvanishes at t = 0 and ends at the sin bt ~ ~ vector J r given by J r . sin br P i What do we really mean by this? Supp ose J r = cer with our frame as ab ove. i sin bt i ~ ~ ~ ~ Let our new eld b e de ned by V t= c r ~e t. Notice that V r =Jr, V 0=0, i sin br cos bt 0 i ~ ~ V t=b c r ~e t since the frame is parallel, and V is orthogonal to ~ at every p oint i sin br 0 ~ since J r is and wehave a parallel frame along a geo desic, soe ~ t= Tt= ~t for all t. 1 Now compute letting c = c r using the Index Inequality 3.2 on the manifold of constant i i 2 curvature b whichwe can do since we are inside the injectivity radius by assumption that r =b: sinbt b b i ~ ~ I J; J I c e~ t i r r sinbr Z h i r X X 1 2 2 i 2 2 2 i 2 = b cos bt c b sin bt c dt 2 sin br 0 X b i 2 = sinbr cos br c 2 sin br = b cotbr g J r ;Jr Wehave just ab out proven an estimate on the metric tensor. We use 4- 1. Wenow know that 0 g J r ;Jr b cotbr g J r ;Jr 4- 2 @ i Nowifwe take J t to b e a Jacobi eld such that J r = j for = for some i, then Q @ 2 0 2 g J r ;Jr = r g kJ 0k by Prop osition 3.5, so we can compute: 0 b cotbr g J r ;Jr g J r ;Jr 0 g J r ;Jr b cotbr g J r ;Jr @ gJ r;Jr @r = 2g J r ;Jr 3 @ 2 0 2 r g kJ 0k @r = 2 0 2 2r g kJ 0k 1 @ p = + log g r @r And thus wehave our rst estimate on the metric tensor in p olar co ordinates: @ p 1 log g b cotbr @r r Notice that 1 sinbr d sinbr d b cotbr log = log = + log b r dr r dr br so if weintegrate from 0 to r we get 0 p sin br 0 g r ; log log 0 br 0 @ @ 2 2 since at 0, the metric is Euclidean, so g 0; = 1 since in R wehaveg ; =r and @ @ sin br lim =1.Thus wehave s!0 br sin br g r; br n1 for any , a co ordinate on S . 5. Lower Bounds on Determinant of Metric. We shall now use what we proved ab ove to prove: 2 Theorem 5.1. If the sectional curvatureofMisboundedabove by b then we have the fol lowing two estimates, provided r<=b: p @ n1 log jg r; j n1b cot br @r r p n1 sin br jg r; j br where jg j = det g i j .
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