Estimates on the Metric Tensor

Home , Normal coordinates

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,

and we will say that P = 0, Q = r , and T t= t.

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

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

0 0

I V; V = [gV t;V t + g RT t;V tT t;V t] dt

where T t= is the tangential vector eld to .

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

, V 0=0, and J r = Vr, along such that both J and V are orthogonal to T t=

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

P P

is singular at X .

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

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

if wehave a Jacobi eld along a geo desic such that J 0 = 0, then we see that in our

geo desic p olar co ordinates wehave Jacobi elds J t= tkJ 0k where kw k foravector

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 .

4. Lower Bounds for the Comp onents of the Metric Tensor. We shall prove the

following:

Theorem 4.1. If the sectional curvatureofMisboundedabove by b then we have the

fol lowing two estimates, provided r<=b:

@ 1

g b cotbr log

@r r

sin br

g r;

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 ,

or g RT; J T; J b .

We furthermore observe that if J is a Jacobi eld, then, by de nition, J t=RT; J T ,

so we get

0 0 00 0

I J;J= [gJ t;J t + g J t;Jt] dt = g J r ;Jr 4- 1

We are going to compare the metric on M with the metric on a space of constant curvature

b .To do this, we need to make a frame fe t; :::; e tg on M by taking an orthonormal basis

1 n

fe ; :::; e g of T M such that e = 0 and then parallel translating it along . Notice

1 n 1

that since is a geo desic and parametrized by arclength, e t= t. We also need to take

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

v ;w are real-valued functions, then

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,

0 0

I J;J = [gJ ;J +gRT; J T; J ] dt

0 0 2

g J ;J b gJ;J dt

~ ~

= 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

What do we really mean by this? Supp ose J r = cer with our frame as ab ove.

sin bt

~ ~ ~ ~

Let our new eld b e de ned by V t= c r ~e t. Notice that V r =Jr, V 0=0,

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

sin br

since J r is and wehave a parallel frame along a geo desic, soe ~ t= Tt= ~t for all t.

Now compute letting c = c r using the Index Inequality 3.2 on the manifold of constant

i i

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

r r

sinbr

h i

X X

2 2 i 2 2 2 i 2

= b cos bt c b sin bt c dt

sin br

i 2

= sinbr cos br c

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

g J r ;Jr b cotbr g J r ;Jr 4- 2

Nowifwe take J t to b e a Jacobi eld such that J r = j for = for some i, then

2 0 2

g J r ;Jr = r g kJ 0k by Prop osition 3.5, so we can compute:

b cotbr g J r ;Jr g J r ;Jr

g J r ;Jr

b cotbr

g J r ;Jr

gJ r;Jr

2g J r ;Jr 3

2 0 2

r g kJ 0k

2 0 2

2r g kJ 0k

1 @

= + 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

sin br

g r ; log log

@ @

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

sin br

g r;

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:

Theorem 5.1. If the sectional curvatureofMisboundedabove by b then we have the

fol lowing two estimates, provided r<=b:

@ n1

log jg r; j n1b cot br

@r r

sin br

jg r; j

where jg j = det g i j .

We rst take a bunch of Jacobi Fields J ; :::; J such that together with T at r they

1 n1

i j

form a basis for T M .Wenow let jg j = det g and supp ose that in co ordinates wehave

Jt= c and compute:

@ 1 @

log jg j = tr log jg j

@r 2 @r

1 @

j k

i j

= tr g g

2 @r

@ gJ r;J r 1

i j

g =

2 @r r

n 1

i j

= g J r ;J rg

n 1

= g J r ;J r

i 4

0 k 0

We get the last equality seen ab oveby letting J r =cJr since the J r form an

i i

orthonormal basis and compute:

i j i j

0 k 2

g J r ;J rg = c r g k j g

i i

2 k i

= r c

i k

2 i

= r c

= g J r ;J r

We can now use 4- 2 and see that

@ n 1

log jg jn1b cot br

@r r

Integrating this from 0 to r we get

n 1 sin br

n 1b cotbr dr = log

r br

sin br

= log

sin br @ @

b ecause lim =1.Now, when r =0, g is the Euclidean metric, where g ; =r ,

r !0

br @ @

so kg 0; k = det I = 1 where I is the n 1 n 1 identity matrix. Thus we

n1 n1

nally get:

sin br

log jg r ; j log

sin br

jg r ; j

6. Upp er Bounds on Comp onents of the Metric Tensor. We shall prove the follow-

ing:

Theorem 6.1. If the sectional curvature of M is boundedbelow by a then we have the

fol lowing two estimates:

1 @

log g a cothar

@r r

sinh ar

g r;

We b egin as ab ove. We know that for a Jacobi eld J along a geo desic wehave

2 2

gRJ;TT; J a ,so gRT; J T; J a .

Let us de ne a vector eld V t along the geo desic .We rst consider a Jacobi eld J

j .We x a frame fe tg along along such that J 0=0,gJ 0;T0 = 0, and J r =

Q i

sinh at

i i

tet. We can now take as our vector eld V t= as ab ove, so J t=c c r e t.

i i

sinh ar 5

~ ~

Notice that V would b e a Jacobi eld in a manifold M of constant curvature a .We also

see immediately that V 0 = 0 and V r =Jr.

The Index Inequality 3.2 gives us

I J;J I V; V

r r

0 0

= [gV ;V +gRT; V T; V ] dt

0 0 2

g V t;V t + a g V; V dt

~ ~

= I V;V

a 2

where I is the index on a manifold of constant curvature a . Note we can use the Index

Inequality since we are within the injectivity radius and hence there are no conjugate p oints.

Now, recall 4-1, which handles the left side. We also want to compute the right side

i i

again letting c = c r :

a 0 0 2

~ ~

I V;V = gV t;V t + a g V; V dt

h i

X X

2 2

2 i 2 2 i 2

= a cosh at c + a sinh at c dt

sinh ar

2 2

i 2

= a cosh at + a sinh at dt c

sinh ar

= cosh ar sinh ar g J r ;Jr

sinh ar

= a cothar g J r ;Jr

Nowwe can follow the same calculation as ab ove to get the following two estimates:

@ 1

log g a cothar

@r r

sinh ar

g r;

References

[1] Thierry Aubin. Some Nonlinear Problems in Riemannian Geometry. Springer-Verlag,

Berlin, Germany, 1998.

[2] Manifred Perdigao do Carmo. Riemannian Geometry. Birkhauser, Boston, MA, 1992. 6