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Notes for Math 760 – Topics in Differential Geometry
January 2015
0 Introduction: Problems in Riemannian Geometry
Undefined terms: Differentiable manifold, Riemannian manifold, homogeneous manifold, incompressible submanifold, stabilizer, conformal class
Obviously differential geometry is a sub-study of differentiable manifold theory, and while geometry has its own internal motivations and problems, to a substantial degree geometry exists to serve topology. In some sense the central project of differential topology is to establish a set of numerical or algebraic invariants that can perfectly distinguish manifolds from one another, as the algebraic invariant a^2 + b^2 = c^2 distinguishes right triangles in Euclidean space from all other triangles (or, more pedantically, the numerical invariants a, b, c (side lengths) distinguish all triangles up to similarity).
Nineteenth century mathematics determined that compact 2-manifolds could be classi- fied by a single numerical invariant: the Euler number (and when χ = 0 also orientability). Part of the cluster of results surrounding this was the geometrization (also known as uni- formization) of compact 2-manifolds: any compact 2-manifold possesses a metric of constant curvature +1, 0, or -1.
Later, the ideas of geometrization leached into other areas of manifold theory; of special note is the case of 3-manifolds. As opposed to the 3 homogeneous geometries of 2-manifolds, Thurston was noticed that 3-manifolds could admit just 8 homogeneous geometries (with compact stabilizers), called the model geometries. He noticed that, although incompressible spheres and tori obstructed the existence of a model geometry, incompressible 2-manifolds of other kinds did not. This, along with other evidence, led Thurston conjecture that any 3-manifold can be cut along incompressible tori (in an essentially unique way) such that the resulting connected components admit a complete metric with one of the 8 geometries.
Since the topological structure of any manifold with a model geometry is completely understood, if any compact 3-manifold admits such a decomposition, then the topology of all compact 3-manifolds would be understood.
Starting in the early 1980’s, Richard Hamilton began a program of using the Ricci Flow to prove the geometrization conjecture for 3-manifolds. The Ricci flow is an evolution equation for the metric:
dgij dt
= −2Ricij. (1)
This is a non-linear 2nd order PDE, which resembles a heat equation, or more accurately a reaction-diffusion equation. Hamilton proved that if singularities do not develop, the resulting manifold in the limit carries a model geometry. Singularities proved difficult to control, but in 2003 Perelman classified the singularities, proved that there are at most finitely many of them, and thereby completed the geometrization program.
The great success of differential-geometric methods in the solution of Thurston’s os- tensibly topological program has spurred development of such methods in other dimensions. To simplify (almost to absurdity!), the program has three parts
In dimension n, determine the “canonical” geometries, and the topological restrictions associated to such a geometry
Determine a canonical way to break a manifold into pieces, each of which can be given a canonical geometry
Determine the topology of the original manifold from the topology of the pieces with their gluing pattern
By-and-large, we the mathematical community is stuck on (1), and we expect to be stuck there for the rest of our lives! (Except in some important special cases.) Surely Einstein metrics should be canonical; then as an example of the difficulty of (1), you may ask which manifolds admit an Einstein metric? If you answer this, even in the case n = 4, you’ll get a fields medal. However we now completely understand which K¨ahler manifolds admit K¨ahler-Einstein metrics.
Many of the problems we’ll explore are involved in the search for canonical metrics. Our list of topics (we’ll surely not get to all of them) is
- The Yamabe Problem: does a conformal class contain a constant-scalr curvature met- ric? Is it unique?
- (prescribed scalar curvature) Given a function f : M n^ → R, is there a metric on M n so that scal = f?
- The Ricci flow: can we canonically smooth-out a metric using a heat-like flow in curvature, like the heat flow “smooths out” any distribution of heat on a compact body? (Possibly related topic: the Yamabe flow.)
- Complex geometry, especially K¨ahler geometry: if we add additional structure, do our problems become more solvable? (Answer: Yes, but they’re still hard.)
Theorem 1.1 (Federer-Fleming)
Iν (Ω) = Cν (Ω)
Proof. Step I: Iν (Ω) ≥ Cν (Ω)
Given any domain Ωo ⊂⊂ Ω, define the function
f(x) = max
dist(x, Ω 0 ), 0
It is easy to prove that lim→ 0
f
ν− 1 ν (^) = |Ω|. It is somewhat more complicated, but still
not difficult to prove that lim→ 0
∇f = |Ωo|. Therefore
|∂Ωo| = lim → 0
|∇f| ≥ Cν (Ω)
f ν−^ ν 1
) ν− ν^1 ≥ Cν (Ω) |Ωo|
ν− ν 1
. (6)
Step II: Cν (Ω) ≥ Iν (Ω)
This is the more interesting case. Assume f is smooth, with isolated critical points. On any non-critical point p, let dA indicate the area measure on the level-set of f passing through p. The co-area formula is
dV olp =
|∇f |
df ∧ dA (7)
Then using Ωt = {|f | ≥ t} to denote the superlevel sets, we have
∫ |∇f | dV ol =
df ∧ dA =
∫ (^) sup |f |
inf |f |
{f =0}
dA df =
0
|∂Ωt| dt
≥ Iν (Ω)
0
|Ωt|
ν− ν 1
and using the “layer-cake” representation and then a change of variables (a l´a multivariable calculus), we have
∫ f
ν ν− (^1) = ν ν − 1
Ω
∫ (^) |f (p)|
0
t
1 ν− (^1) dt dV ol(p) = ν ν − 1
0
Ωt
t
1 ν− (^1) dV ol dt
ν ν − 1
0
t ν−^11 |Ωt| dt
Lastly we have to connect these. We have
Lemma 1.2 If g(t) is non-negative and decreasing, and if s ≥ 1 , then
( s
0
ts−^1 g(t) dt
) (^1) s ≤
0
g(t)
1 s (^) dt (10)
Proof. First taking a derivative
d dT
s
∫ T
0
ts−^1 g(t) dt
) (^1) s = T s−^1 g(T )
s
∫ T
0
ts−^1 g(t) dt
) 1 −ss
≤ T s−^1 g(T )
(^1) s
s
∫ T
0
ts−^1
) 1 −ss = g(T )
(^1) s
and then taking an integral, we get
( s
0
ts−^1 g(t)
) (^1) s ≤
0
g(t)
(^1) s dt. (12)
Applying this lemma with s = (^) νν− 1 we get
(∫ f
ν− 1 ν
) (^) ν−ν 1
ν ν − 1
0
t ν−^11 |Ωt| dt
) ν− ν^1 ≤
0
|Ωt|
ν− 1 ν (^) ≤ 1 Iν (Ω)
|∇f | (13)
giving the inequality Iν (Ω) ≤ Cν (Ω).
Theorem 1.3 For any p < ν we obtain embeddings L
pν ν−p (^) ⊂ W 01 ,p.
The critical value of ν is ν = n. The Sobolev constant on standard Euclidean space has Cn(Rn) nonzero, but for any other value of ν, Cν (Rn) = 0.
If f has more than one weak derivative, then the embedding theorem also improves.
Theorem 1.4 If 1 ≤ p < ∞, k ∈ N, and kp < ν then we obtain embeddings L ν−pνpk ⊂ W 0 k,p.
So what happens if pk ≥ ν? As pk ↗ ν, then (^) νpν−pk ↗ ∞, so perhaps the embedding is into L∞. This naive supposition fails because the constant in the Sobolev inequality also degenerates (see the exercises).
However if pk > n, then we obtain strong consequences indeed.
Theorem 1.5 If l + α = k − np and Cn(Ω) > 0 , then
W 0 k,p (Ω) ⊂ C 0 l,α (Ω) (14)
Our last word is the Kondrachov-Rellich embedding theorems:
The tensor ∇T has an expression in coordinates, where, by convention, the derivative is indicated with a comma:
∇T = Ti 1 ...ip^ j^1 ...jq^ ,k dxi^1 ⊗... dxip^ ⊗
∂xj^1
∂xj^1
⊗ dxk. (18)
One can compute the following expression for the component functions:
Ti 1 ...ip^ j^1 ...jq^ ,k = d dxk^
Ti 1 ...ip^ j^1 ...jq^ + Γski 1 Ts...ip^ j^1 ...jq^ +... + Γskip Ti 1 ...sj^1 ...jq
- Γj ks^1 Ti 1 ...ss...jq^ +... + Γj kspTi 1 ...sj^1 ...s.
Obviously this can be iterated: ∇T , ∇^2 T , ∇^3 T etc. The so-called rough Laplacian of a tensor T is the trace of the Hessian:
4 T = gklTi 1 ...ip^ j^1 ...jq^ ,kl dxi^1 ⊗... dxip^ ⊗
∂xj^1
∂xj^1
In particular, if f is a function, then
4 f = gij^ ∂^2 f ∂xi∂xj^
− Γk^ df dxk^
, Γk^ , gij^ Γkij. (21)
1.4.2 Forms, the Hodge Duality Operator, and the Hodge Laplacian
We recall a few definitions. We have the sections of k-forms, denoted Γ
k (^) M n
. By abuse
of notation, we denote this by simply
∧k M n^ or just
∧k
. The Hodge duality operator or just Hodge star is term given to the tensoral forms of the volume form:
dV ol =
gij dx^1 ∧ · · · ∧ dxn, ∗ :
∧ (^) k −→
∧ (^) n−k (22)
The ∗ operator is an isomorphism, as ∗∗ = (−1)k(n−k). It is also positive definite in the sense that ∗ (η ∧ ∗η) ≥ 0 with equality if and only if η = 0. Polarizing this quadratic operator, we define an inner product
〈η, γ〉 = ∗ (η ∧ ∗γ). (23)
If η, γ ∈
∧k then η ∧ ∗γ ∈
∧n , so we can integrate it. This gives rise to an L^2 inner product
(η, γ) ,
η ∧ ∗γ. (24)
Now let’s look at the exterior derivative d :
∧k →
∧k+
. We would like to find its L^2 adjoint. Now d is a linear operator, but is only densely defined. If we restrict the L^2 space to just the C∞^ k-forms, then d is defined everywhere, but it is not a bounded (or continuous)
linear operator, and the inner peoduct space is not a Hilbert space (due to non-completeness of C∞^ under the L^2 norm). Despite all this, we can find an adjoint for d, by making use of Stoke’s theorem. We find that d∗, defined implicitly by
(dη, γ) = (η, d∗γ). (25)
One use of Stokes theorem along with the fact that ∗∗ = (−1)k(n−k)^ gives us d∗^ :
∧k →
∧k− 1
is d∗η = (−1)nk+k+1^ ∗ (d (∗η)).
The first order operators d, d∗^ can be combined give a second order operator, the Hodge Laplacian, by
(^4) H :
k (^) −→
k,
(^4) H η = d∗dη + dd∗η.
This is not the same as the rough Laplacian, although at the first and second order levels they are essentially the same. Notice that 4 is negative definite and (^4) H is positive definite:
( 4 η, η) = −|∇η|^2 ( (^4) H η, η) = |dη|^2 + |d∗η|^2.
It is possible to prove that we have Bochner formulas for the Laplacians
(^4) H η = −4η + F (g, η) (28)
where F is a zero-order differential operator (a linear operator) in η, and a second-order non-linear operator in g. The following commutation relations are also often very useful:
[d, (^4) H ] = d (^4) H − 4H d = 0 [d∗, (^4) H ] = d∗ (^4) H − 4H d∗^ = 0 [∗, (^4) H ] = ∗4H − 4H ∗ = 0.
There are no such simple commutation relation for the rough Laplacian, although later we shall talk about the important commutator [ 4 , ∇].
1.5 Suggested Problems for 1/
Prove formulae (19) and (21).
If f is a function, prove the classic formula 4 f = √det^1 g∂x∂i
det g gij ∂f∂xj
- Prove ∗∗ :
∧k →
∧k is the operator (−1)k(n−k).
- Prove the codifferential operator is d∗^ = (−1)nk+k+1^ ∗ d∗ :
∧k →
∧k− 1 .
- Prove that the inner product 〈η, γ〉 is symmetric (hint: work on a basis).
where g ∈ W 01 ,^2 (Ω). Using our “reverse Sobolev inequality” along with the actual Sobolev inequality gives
Cn(supp η)^2
η^2 γ^ f pγ
) (^) γ^1 ≤
|∇(ηf
p (^2) )|^2
|∇η|^2 f p^ +
p^2 2
η^2 f p−^2 |∇f |^2
p p − 1
|∇η|^2 f p^ +
p^2 p − 1
η^2 f pu
With H¨older’s inequality we obtain
Cn(supp η)^2
η^2 γ^ f pγ
) (^1) γ
p p − 1
|∇η|^2 f p^ + p^2 p − 1
η^2 γ^ f pγ
) (^1) γ (∫
supp η
u n 2
) (^) n 2
[
Cn(supp η)^2 −
p^2 p − 1
supp η
u
n 2
) (^) n^2 ] (∫ η^2 γ^ f pγ
) (^1) γ ≤ 2
p p − 1
|∇η|^2 f p
Thus, provided
supp η U^
n 2 is sufficiently small with respect to both p and the Sobolev constant, then the local Lγp-norm of f is controlled in terms of the local Lp-norm of f. Lifting apriori from lower Lp-norms to higher Lp-norms is called bootstrapping.
Lemma 1.7 (First Bootstrapping Inequality) Assume 4 f + uf ≥ 0 , f ≥ 0 , f ∈ Lploc
for some p > 1 , u ∈ L
n 2 loc, and^ η^ ∈^ C
∞ c. Then ∫
supp η
u n 2 ≤ p − 1 2 p^2
Cn(supp η)^2 (36)
implies
(∫ η^2 γ^ f pγ
) (^) pγ^1 ≤
4 p^2 p − 1
Cn(supp η)^2
) (^1) p (∫ |∇η|^2 f p
) (^1) p
. (37)
Iterating this estimate as many times as necessary, and choosing appropriate test func- tions at each stage (see below), gives us the following theorem
Theorem 1.8 (First -Regularity Theorem) Assume 4 f + f u ≥ 0 , f ≥ 0 , f ∈ Lploc
for some p > 1 , u ∈ L
n 2 loc, and^ q^ is any number^ q^ ∈^ (p,^ ∞).^ Then there exist constants = (p, q, Cn(Br )) > 0 and C = C(p, q, Cn(Br ), r) < ∞ so that ∫
Br
u
n 2 ≤ (38)
implies (∫
Br/ 2
f q
) (^1) q ≤ C
Br
f p
) (^1) p (39)
In other words, f ∈ Lploc, p > 1 and u ∈ L
n 2 loc implies^ f^ ∈^ L
q loc for all^ q^ ∈^ (p,^ ∞). However both = (p, q, Cn) and C = C(p, q, Cn, r) deteriorate as q → ∞ (see the exercises), so we do not obtain apriori L∞^ bounds.
1.6.2 Cutoff Functions
It is worthwhile to formalize the choice of test functions necessary to the proof of Theorem 1.8. The test functions we construct below are called cutoff functions.
Let m ∈ M n^ be any point, and let r = dist(m, ·) be the distance function from m. Distance functions are not smooth, but are Lipschitz and therefore weakly differentiable. Since we encounter at worst gradients of η in our integral estimates, the fact that our “test functions” are just C^0 ,^1 and not C∞^ is no bother to us.
Given some radius r 0 ∈ (0, ∞), we may wish to have a test function that is zero outside Br 0 (m), is 1 inside Br 0 / 2 (m) and has bounded gradient in the intermediate annulus Br 0 (m) \ Br 0 / 2 (m). Then define
η(r) =
1 , r ∈ [0, r 0 /2] 2 − (^) r^20 r , r ∈ (r 0 / 2 , r 0 ) 0 , r ∈ [r 0 , ∞)
We have ∇η = η′∇r and so this function is Lipschitz, and |∇η| = |η′| ≤ (^) r^20 (a.e.), as the gradient of a distance functions has norm 1 (a.e.).
It is also sometimes necessary to have a variety of cutoff functions, with the collection of cutoff functions operating on progressively smaller nested balls, say. To this end consider the functions
ηi(r) =
1 , r ∈ [0, (2−^1 + 2−i−^2 )r 0 ] 1 − 2
i+ r 0
r − (2−^1 − 2 −i−^2 )r 0
, r ∈
(2−^1 + 2−i−^2 )r 0 , (2−^1 + 2−i−^1 )r 0
0 , r ∈ [(2−^1 + 2−i−^1 )r 0 , ∞).
The gradients are
|∇ηi| =
0 , r ∈ [0, (2−^1 + 2−i−^2 )r 0 ] 2 i+ r 0 , r^ ∈^
(2−^1 + 2−i−^2 )r 0 , (2−^1 + 2−i−^1 )r 0
0 , r ∈ [(2−^1 + 2−i−^1 )r 0 , ∞).
Notice that the supports supp|∇i| are non-overlapping.
This can be iterated all the way down, to obtain
(∫
B(2− (^1) +2−i− (^2) )r 0
f pi+
) (^) p 1 i+ ≤
[ (^) i ∏
k=
C(n) p^1 k (^) C p^2 nk
[
(2kr 0 )^2 + p
n 2 k
] (^) p 1 k
] (∫
Br 0
f p
) (^1) p
. (47)
Theorem 1.10 (Epsilon-Regularity) Suppose f ≥ 0 weakly satisfies 4 f + uf ≥ 0 ,
and assume f ∈ L^2 loc and u ∈ L
n 2 γ loc.^ Given any radius^ r^0 >^0 and point^ m^ ∈^ M^ n, and
abbreviating Cn(Br 0 (m)) by Cn, there exist constants = (n, Cn) and C = C(n, Cn, r 0 ) so that if ∫
Br 0 (m)
u n 2 γ ≤ (48)
then
sup Br 0 / 2 (m)
|f | ≤ C
Br 0 (m)
f 2
Proof. One simply checks that [ (^) i ∏
k=
C(n) p^1 k (^) C p^2 nk
[
(2kr 0 )^2 + p
n 2 k
] (^) p^1 k
]
[ (^) i ∏
k=
C(n)
1 pγk^ C
2 n^ pγk^ [(2kr 0 )^2 +^ p^ n^2 γk^ n^2 ]^ pγ^1 k
]
is uniformly bounded independently of i (see exercises). Setting p = 2 and taking i → ∞ on both sides of (47) gives the result.
This method of finding apriori L∞^ control after assuming just L^2 control is called Moser iteration or Nash-Moser iteration, depending on whether you’re an analyst or a geometric analyst.
1.7 Suggested Problems for 1/
Supply a proof for Theorem 1.8. Give explicit estimates for and C in terms of p, q, Cn(Br ), and r.
By considering functions f : Rn^ → R of the kind f = r−k^ (log r)−l, show that there exist functions f , u that satisfy the hypotheses of Theorem 1.8 but so that f 6 ∈ L∞ loc. Conclude that Theorem 1.8 is optimal. (Hint: computing Laplacians should be easy after you compute 4 r and verify the identity 4 f (r) = f ′′(r) + f ′(r) 4 r.)
Explicitly estimate
k=0 C(n)^
1 γk^ C
2 n^ γk^ [(2kr 0 )^2 +^ γk^ n^2 ]^ γ^1 k^ in terms of^ n,^ Cn and^ r 0 , thereby completing the proof of Theorem 1.10. You could also try leaving p > 1 and estimating the product in terms of n, Cn, r 0 , and p, thereby bounding |f | in terms of any
f p
) (^1) p , p > 1, instead of just
f 2
1.8 Variational Problems and Euler-Lagrange Equations
1.8.1 Basic Quadratic Functionals and Linear Euler-Lagrange Equations
Let (M n, g) be a Riemannian manifold, which needn’t be compact. If T is any W 1 ,^2 tensor, consider the quadratic functional
F(T ) =
M n
|∇T |^2 dV ol. (51)
One is sometimes interested in finding minimizers (or more generally extremizers) of such a functional. To do this, we consider variations Tt = T + tH of the tensor T and compute the first order effect on F (when M n^ is non-compact it is normal to put a restriction on H, namely that H ∈ W 01 ,^2 or H ∈ C c∞ even, in order that we can use Stokes’ theorem). If first derivatives of F(Tt) vanish regardless of what variation H is chosen, then the tensor T is said to be a critical point for F. Then if T is an extremizer, then we compute
d dt
t=
F(Tt) =
d dt
|∇Tt|^2
〈∇T, ∇H〉 = − 2
〈4T, H〉.
Since H is arbitrary, we can choose H = η 4 T , where η ≥ 0 is any test function, to obtain the requirement that 4 T = 0. One easily sees that indeed 4 T = 0 if and only if T extremizes F. In general, the pointwise equations one obtains as a requirement for a quantity to extremize a functional are called the Euler-Lagrange equations of that functional.
There is a more abstract way to look at this process. Let T be the vector space of all W 1 ,^2 sections of tensors on M n; this is a Banach space. Then F : T −→ R is a map from one Banach space to another, and more specifically, it is just a function on the (infinite dimensional) space T. As such, we should be able to linearize it (that is, find its “gradient”). This linearization at T ∈ T is called its Frechet derivative of F at T , and is sometimes denoted
DF
T :^ W^
1 , 2 0 −→^ R.^ (53)
The first derivative of F at T in the “direction” of H is
DF
T (H) =^
d dt
t=
F(T + tH). (54)
We are looking for those points T ∈ T at which the linearized operator DF|T vanishes. Unfortunately we won’t have the time to explore this abstract point of view much further than this.
Obviously functionals for other curvature quantities also exists:
RM(M n, g) =
M n
| Rm |
n (^2) dV ol
RIC(M n, g) =
M n
| Ric |
n 2 dV ol
W(M n, g) =
M n
|W |
n 2 dV ol.
In dimension 4 these all take on special interest, as n 2 = 2. We obtain scale-invariant quadratic curvature functionals ∫
M n
| Rm |^2 dV ol,
M n
| Ric |^2 dV ol,
M n
| Rı◦c |^2 dV ol ∫
M n
R^2 dV ol,
M n
|W |^2 dV ol,
M n
|W ±|^2 dV ol.
1.8.4 Riemannian Variational Formulas and non-linear Euler-Lagrange Equa- tions
To compute variations of these functionals, we have to compute variations of the usual Riemannian quantities. Suppose we vary the metric
gt = g + th (63)
where h = hij dxi^ ⊗ dxj^ is any symmetric 2-tensor. We have dgdt = h. With the usual formula
Γkij = (^12)
∂gis ∂xj^ +^
∂gjs ∂xi^ −^
∂gij ∂xs
gsk^ we compute
dΓkij dt
t=
∂his ∂xj^
∂hjs ∂xi^
∂hij ∂xs
gsk^ −
∂gis ∂xj^
∂gjs ∂xi^
∂gij ∂xs
gsugvkhsv. (64)
For theoretical reasons we know that Γ˙kij is tensorial, so we apply the following standard
trick: express Γ˙kij in a coordinate system in which all Christoffel symbols vanish, then convert all partial derivatives to covariant derivatives. In geodesic normal coordinates, at any single chosen point p we have gij = δij + O(2) so the ∂g∂x terms vanish. Thus we have
dΓkij dt
t=
∂his ∂xj^
∂hjs ∂xi^
∂hij ∂xs
gsk
dΓkij dt
t=
(his,j + hjs,i + hij,s) gsk
Using the standard formulas for Rmijk l^ in terms of the Christoffel symbols, we obtain
d dt
t=
Rm (^) ijkl(gt) =
(hjs,ki + hks,ji − hjk,si − his,kj + hks,ij − hik,sj ) gsl
d dt
t=
Ric (^) ij (gt) =
(hsi,js + hsj,is − hjk,ss^ − (T r h),ij )
d dt
t=
R(gt) = hst,ts − 4(T r h) − Ricij hij
From this, along with (^) dtd dV ol = 12 (T r h) dV ol we can easily compute
d dt
t=
R dV ol =
Ric −
R g , h
dV ol (67)
so that a stable point for the Hilbert functional occurs when the gravitational tensor van- ishes:
Gij , Ric (^) ij −
Rgij
Gij = 0.
1.9 Suggested Problems for 2/
- (Other Elliptic Equations) If (M n, g) is a compact Riemannian manifold and f, h are integrable functions, determine the Euler-Lagrange equations for the quadratic functional
F(u) =
f |∇u|^2 + h|u|^2. (69)
- (The p-Laplacian) Determine the Euler-Lagrange equations for the non-quadratic func- tionals
F(u) =
|∇u|p^ (70)
where p > 1 and, as usual, we take (M n, g) to be compact. Find the Euler-Lagrange equations for this functionals; these are known as the p-Laplace equations. Obviously the 2-Laplacian is just the Laplacian. In the limiting cases p = 1 and p = ∞, what are reasonable notions of the 1- and ∞-Laplacians?
- Consider the Riemannian metric on R^2
gij =
(1 + r^2 )^2 δij. (71)
This pulls back to S^2 , under standard streographic projection, to the metric of constant curvature +1. Given any constant Λ > 0, we have diffeomorphisms R^2 → R^2 given
1.10.1 Derivative Commutator Formulas
From basic Riemannian geometry we have the definition of the Riemann tensor as a com- mutator: if X is any vector field then
Xl,ij − Xl,ji = RmijslXs. (72)
By computation this can be extended to other tensors:
T l^1 ...ls^ ,ij − T l^1 ...ls^ ,ji = Rmijsk^1 T sl^2 ...ln^ + Rmijsl^2 T l^1 s...ln^ +... + Rmijsln^ T l^1 ...ln−^1 s^ (73)
and to covectors η = ηidxi
ηk,ij − ηk,ji = −Rmijksηs Tk 1 ...kn,ij − ηk,ji = − Rmijk 1 sTsk 2 ...kn − Rmijk 2 sTk 1 s...kn −... − Rmijkn^ sTk 1 ...kn− 1 s.
This can be extended to tensors of any type. For instance
Tabc,ij − Tabc,ji = RmijasTsbc^ − RmijsbTasc^ − RmijscTabs^ (75)
1.10.2 The Second Bianchi Identity and the Riemann Tensor
Both the first and second Bianchi identities can be derived from the diffeomorphism invari- ance of the Riemann tensor (Reference: J. Kazdan, 1981). If we regard Rm = Rm(g) as a second order differential operator on the metric g, then diffeomorphism invariance takes the form
Rm(ϕ∗g) = ϕ∗(Rm(g)) (76)
where ϕ : M n^ → M n^ is any diffeomorphism and ϕ∗^ is the pullback. Letting ϕt be a time- dependent family of diffeomorphisms with a well-chosen variational field, then taking a Lie derivative on both sides of (76) gives the second Bianchi identity (see exercises). A similar process gives the first Bianchi identity. In this light the Bianchi identities are understood as Leibniz rules.
Alternatively, the second Bianchi identity identity can be computed directly from the definition of Rm, where it ultimately follows from the Jacobi idenity, which itself is both an expression of diffeomorphism invariance (of the bracket), and also a Leibniz rule.
The 2nd Bianchi identity can be expressed in either of the following two ways: Rmijkl,m + Rmijlm,k + Rmijmk,l = 0 Rmijkl,m + Rmjmkl,i + Rmmikl,j = 0.
Tracing once gives
Ricij,k − Ricik,j = Rmsijk,s^ (78)
Tracing again gives
R,k = 2Ricsk,s. (79)
1.10.3 The Laplacian of the Riemann Tensor
Combining the 2nd Bianchi identity with the commutator formula, we can obtain and ex- pression for the Laplacian of the full Riemann tensor. First using the 2nd Bianchi idenitity we get
( 4 Rm)ijkl , gstRmijkl,st = −gstRmijls,kt − gstRmijsk,lt
Using the commutator formula on both terms gives
( 4 Rm)ijkl = −gstRmijls,tk − gstRmijsk,tl
where “... ” indicates a further seven quadratic monomials in the Riemann tensor. Whenever the exact expression doesn’t matter, we shall abbreviate any linear combination of tensor products and traces of a tensor T with a tensor S simply by S ∗ T. To (81), apply the 2nd Bianchi identity again to get
( 4 Rm)ijkl = gstRmjtls,ik + gstRmtils,jk + gstRmjtsk,il + gstRmtisk,jl + Rm ∗ Rm = −Ricjl,ik + Ricil,jk + Ricjk,il − Ricik,jl + Rm ∗ Rm
The second derivative terms are now entirely on various copies of the Ricci tensor. Schemat- ically expressed, we have
4 Rm = ∇^2 Ric + Rm ∗ Rm. (83)
In the n ≥ 3 Einstein case we have Ric = λg where λ = const, and so we obtain the tensoral elliptic equation 4 Rm = Rm ∗ Rm. This is difficult to deal with using the above elliptic theory, however, because it was developed for functional Laplace equations of the sort 4 f ≥ uf. We therefore resort to the so-called Kato inequality: for any tensor T we have
|∇|T || ≤ |∇T |. (84)
Then the two identities 1 2
4|T |^2 = 〈4T, T 〉 + |∇T |^2
4|T |^2 = |T |4|T | + |∇|T ||^2
along with the Kato inequality give
|T |4|T | ≥ 〈4T, T 〉. (86)
Using T = Rm and | 〈Rm ∗ Rm, Rm〉 | ≥ −C| Rm |^3 (from Cauchy-Schwartz) we get the equation
4| Rm | ≥ C(n)| Rm |^2 (87)
which holds strongly away from | Rm | = 0 and holds weakly everywhere.
