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The concepts of torsion and curvature in Riemannian geometry. It covers the definition of the torsion tensor, the torsion-free condition, the relationship between the torsion and the connection matrix, and the metric compatibility condition. The document also discusses the curvature tensor, its symmetries, and its relation to the Levi-Civita connection. Furthermore, it touches upon the topic of semi-Riemannian manifolds and their curvature.
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Department of Mathematics Geometry of Manifolds, 18. Spring 2005 Lecture Notes
In this lecture we introduce the most important connection. This is the Levi-Civita connection in the tangent bundle of a Riemannian manifold.
When dealing with tensors on a manifold it is convient to use the following conventions. When we choose a local frame for the tangent bundle we write e 1 ,... en for this basis. We always index bases of the tangent bundle with indices down. We write then a typical tangent vector
∑^ n
i=
Xiei.
Einstein’s convention says that when we see indices both up and down we assume that we are summing over them so he would write
X = Xiei
while a one form would be written as
θ = aiei
where ei^ is the dual co-frame field. For example when we have coordinates x^1 , x^2 ,... , xn^ then we get a basis for the tangent bundle
∂/∂x^1 ,... , ∂/∂xn
More generally a typical tensor would be written as
T = T ijklei ⊗ ej^ ⊗ ek^ ⊗ el
Note that in general unless the tensor has some extra symmetries the order of the indices matters. The lower indices indicate that under a change of frame fi = Cij^ ej a lower index changes the same way and is called covariant while an upper index changes by the inverse matrix. For example the dual coframe field to the fi, called f i^ is given by
f i^ = Dij ej
where Dij is the inverse matrix to Cj i (so that Dij Cj k = δik.) The compo- nents of the tensor T above in the fi basis are thus
T ijkl^ = T i
′ j′k′^ l′ Di i′^ C j′ j C k′ kD l l′
Notice that of course summing over a repeated upper and lower index results in a quantity that is independent of any choices. Given a vector bundle over our manifold which is not the tangent bundle or tensors on the tangent bundle we use a distinct set of indices to indicate tensors with values on that bundle. If V → M is a vector bundle of rank k with a local frame vα, 1 ≤ α ≤ r we would write
s = cαi vα ⊗ dxi
for a typical section of the bundle T ∗M ⊗ V Given a ∇ connection in V we write
∇s = sα;idxi^ ⊗ vα.
That is we think ∇s as a section of T ∗M ⊗ V as opposed to the possibly more natural V ⊗ T ∗M. Or more concretely the semi-colon is also indicating that the indices following the semi-colon are to really be thought of comming first and the opposite order. Our convention here is designed to be more consistent with the mathematical literature. In the physics literature for example “Graviation” by Misner, Throne and Wheeler. So for a connection in the tangent bundle if we have a vector field with components Xi^ we have write Xi,j for the components of its covariant derivative. The Christoffel
Proof. Notice that there are n^3 distinct functions involved in defining a con- nection. Recall that the torsion free condition implies n^2 (n − 1)/2 relations amongst these functions: Γkij = Γkji. (1)
The metric compatibility implies n^2 (n + 1)/2 relations amongst these func- tions:
gij,l = 〈∇ ∂ ∂xl
∂xi^
∂xj^
∂xi^
∂xl
∂xj^
= gkj Γkli + gikΓklj.
Hence using equation 1
gjl,i + gil,j − gij,l = gklΓkij + gjkΓkil +gkj Γkji + gikΓkjl −gkj Γkli − gikΓklj = 2 gklΓkij
and the result follows.
We will now redo the existence of the Levi-Civita connection from the point of view of the induced connection on the cotangent bundle. This method is more compuationally effective. A connection ∇ in the tangent bundle induces a connection ∇∗^ in the cotangent bundle by the requiring
Xθ(Y ) = ∇∗ X θ(Y ) + θ(∇X Y ).
After this section we’ll drop the ∗ from the notation nice the meaning will be clear from context. To justify this definition think about the condition that the parallet transport for the two connections is equivalent. Thus if Γkij are the Christoffel symbols for ∇ we have
∂xi
dxk^ = −Γkij dxj^.
Lemma 1.2. The condition that ∇ is torsion free is equivalent to the con- dition that for any one-form α we have
dα(X, Y ) = ∇∗ X α(Y ) − ∇∗ Y α(X).
Proof. To see this notice
∇∗ X α(Y )−∇∗ Y α(X) − dα(X, Y ) = Xα(Y ) − α(∇X Y ) − Y α(X) + α(∇Y X) − (Xα(Y ) − Y α(X) − α([X, Y ])) = α(−∇X Y + ∇X Y + [X, Y ]).
A crucial step is to use an orthonormal frame e 1 ,... , en and dual coframe e^1 , e^2 ,... , en^ then we have the Christoffel symbols for the connection
∇ei ej = Γkij ek
∇∗ ei ek^ = −Γkij ej
Thus connection matrix for the dual connection is
∇ek^ = −Γkij ei^ ⊗ ej
The torsion free condition then implies
dek^ = −Γkij ei^ ∧ ej
From the equation ei〈ej^ , ek〉 = 0
we derive that if ∇ is metric compatible then
Γkij = −Γjik.
Let’s prove, from this point of view, the basic uniqueness theorem the Levi- Civita connection.
Lemma 1.3. There is a unique torsion free metric compatible connection.
with the metric
ds^2 =
x^20
((dx^0 )^2 + (dx^1 )^2 ) +... + (dxn)^2 )
we have
ei^ =
dxi x^0
and hence de^0 = 0 while for i > 0 we have
dei^ = −
dx^0 ∧ dxi (x^0 )^2
= −e^0 ei.
So for i > 0
Ai 0 i =
= −Aii 0.
Thus θi 0 = (Aij 0 − A^0 ji − Aji 0 )ej^ = −ei
and θij = 0 if i, j > 0.
The calculations above are straighforward but it is natural to look for a home for them. The bundle of orthonormal frames provides the right framework. After all in particular in the method of moving frames our choice was a local orthonormal frame field. A better way to view what is going on is to work with all frames simulatenously. The primordial object is the bunlde of frames (bases) of the tangent bundle π : Fr(M ) → M. Given a frame f 1 ,... , fn and an matrix aij we get a new orthonormal frame f (^) i′ =
∑n j=1 aij^ fj^ or a little more clearly we think of the orginal frame f = {f 1 ,... , fn} as giving an isomorphism
f : Rn^ → TxM.
Then an isomorphism of a : Rn^ → Rn^ we get a new frame
f ◦ a : Rn^ → TxM.
This make Fr(M ) into a principal Gl(n) bundle. That is there is a map
Fr(M ) × Gl(n) → Fr(M )
so that (f A)B = f (AB).
Each orbit of the action is precisely one fiber of the projection and the action is effective (f A = f =⇒ A = 1). Informally it is a bundle of groups where the transition functions act by left multiplicition on the group leaving the right action to act on the resulting bundle. A connection gives rise to an Gl(n)-invariant horizontal subbundle of the tangent bundle of Fr(M ). By this we mean a subbundle H ⊂ T Fr(M ) so that π∗|H : H → π∗T M is an isomorphism and so that for all a ∈ Gl(n) we have (Ra)∗H = H. The horizontal bundle associated to a connection is the lift of the tangent space given by the connection. There is an exact sequence
0 → V T Fr(M ) → T Fr(M ) → π∗(T M ) → 0.
Here V T Fr(M ) denotes the “vertical tangent bundle” to FrOr(M ) i.e. the kernel of dπ. The vertical tangent bundle is trivial and each fiber is isomor- phic to the Lie algebra of Gl(n) as follows. Let A be a matrix and consider the path e exp(tA). The derivative of the path at t = 0 defines a map
ιe : gl(n) → V TeFr(M ).
We can thus rewrite this exact sequence as
0 → P × gl → T Fr(M ) → π∗(T M ) → 0.
A connection is then a splitting of this exact sequence
0 → P × gln →^ ← T Fr(M ) → π∗(T M ) → 0.
Let us work this out explictly. Given a connection in a vector bundle if we choose local coordinates xi^ in the base and a local frame e = (eα). The frame e gives us a local trivialization so the principal bundle becomes
U × Gln
We get Christoffel symbols
∇ ∂ ∂xi^ eα = Γβiαeβ.
and we pull back Note that if ˜γ(t) is the horizontal lift at a frame e of γ then ˜γ(t)A is the horizontal lift of γ at eA. This implies that right translation preserves the horizontal space of the connection and in particular that
θeA = R∗ Aθe.
Note that that a section of the tangent bundle is given as a map
X : Fr(M ) → Rn
with the equivariance property
X(f A) = A−^1 X(f )
Given a connection we can This bundle carries a tautological Rn-valued one form
θ : T Fr(M ) → Rn.
It is defined as follows. Given a tangent vector v to T Fr(M ) at a frame f we can project v to T M and expand it in terms of f
θ(v) = f −^1 (π∗(v)).
The tautological one form gives a geometric interpretation of torsion. Under the action of Gl(n) this one-form transforms by
(Ra)∗θ = a−^1 θ
Since θ has values in vector space To this end we set π : FrOr(M ) → M to be the bundle of orthonormal frames on M. This is principal O(n) bundle. Given a frame e 1 ,... , en and an orthogonal matrix aij we get a new orthonormal frame fi =
∑n j=1 aij^ ej^ or a little more clearly we think of the orginal frame e = {e 1 ,... , en} as giving an isometry e : Rn^ → TxM.
Then an isometry of a : Rn^ → Rn^ we get a new frame
e ◦ a : Rn^ → TxM.
Given a connection its parallel transport gives rise to a lift through any point e of the total space of each path in a the base passing through π(e). Taking derivatives we get a lifting of the TeM to TeFrOr(M ) for each e. Notice that the tangent bundle of FrOr(M ) is trivial. As have seen that Given a tangent vector ei in the base its lift to FrOr(M ) at e is e Consider more generally a principal G bundle P → M where G is a Lie group. We will always think of G as acting on P on the right. The vertical tangent bundle of any principal bundle is alway trival and indeed each fiber can be identified with the Lie Algebra of the structure group. To see this let ξ ∈ g be an element of the structure group which we always identify with left invariant vector fields on G.
The curvature of the connection is
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z.
The coordinate expression of R is
∂xi^
∂xj^
∂xk^
= Rijkl^
∂xl^
In the trivialization given by our orthonormal frame the curvature of the Levi-Civita connection has the form
Ωlk = dθlk + θlm ∧ θmk.
This is related to the curvature tensor R as follows. Defining as above
Rijkl = 〈R(ei, ej )ek, el〉.
then we have
Lemma 1.4.
[Ωlk] = [
Rijklei^ ∧ ej^ ].
There is another differential equation obeyed by the curvature called the second Bianchi identity.
Lemma 1.6. dΩlk + θml ∧ Ωmk − Ωmk ∧ θlm = 0
Proof. Taking d of the formula Ωlk = dθlk + θlm ∧ θmk yields
dΩlk = dθlm ∧ θkm − θlm ∧ dθkm = (Ωlm − θln ∧ θmn) ∧ θmk − θlm ∧ (Ωmk − θnm ∧ θkn ) = Ωlm ∧ θmk − θlm ∧ Ωmk
In terms of the curvature tensor R the second Bianchi identity takes the form (∇X R)(Y, Z)W + (∇Y R)(Z, X)W + (∇Z R)(X, Y )W = 0
The sectional curvature is the function on the Grassmanian of two planes given by K(Π) = R(e, f, e, f )
where (e, f ) is an orthonormal frame for Π. More generally if X, Y span Π then
K(Π) =
The sectional curvature determines the full curvature.
∂s∂t
(R(X+tZ, Y +sW, X+tZ, Y +sW )−R(X+tW, Y +sZ, X+tW, Y +sZ)) = R(X, Y, Z, W )
In particular if K is a constant on the fibers of the Grassman bundle we have R(
A Lie group is a differentiable manifold and a group for which the group laws are differentiable. If G is a Lie group then for any g ∈ G we have diffeomorphisms Lg, Rg : G → G
given by left and right multiplication respectively. Thus
(Lg)∗TeG → TgG
is an isomorphism and we can trivialize the tangent bundle of G. The formal definition of the Lie algebra of a Lie group is to use this isomorphism to identify the tangent space at the identity with the left-invariant vector fields.
Definition 1.7. A one-parameter subgroup is a map
γ : R → G
so that γ(t + s) = γ(t)γ(s).
Every ξ ∈ g is tangent to a one pararmeter subgroup. To see this let Ft : U ⊂ R × G → G be the flow for the vector field ξ where U is define so that for each g ∈ G U ∩ R × {g} is the maximal interval of definition of the flow. First of all notice that since ξ is left invariant we have Lg ◦ F is also the flow for ξ from which it follows easily that U = R × G and that
Lg ◦ F (t, h) = F (t, gh) (3)
The flow has the semi-group property,
F (t + s, g) = F (t, F (s, g)) (4)
so if g = F (t, e) then
γ(t + s) = F (t + s, e) = F (s, F (t, e)) = F (s, ge) = gF (s, e) = γ(t)γ(s).
Proposition 1.8. Every compact Lie groups admits a bi-invariant metric.
Proposition 1.9. Let 〈, 〉 be a bi-invariant metric. Then for all left-invariant vector fieldsX, Y, Z we have
For example if M = Rn+1^ and N = Sn^ then
II(X, Y ) = 〈X, Y 〉.
Also notice that there is the following nice formula
Lemma 1.11. If ν is a normal vector field to N and X and Y are tangent then: 〈∇X ν, Y 〉 = −〈ν, II(X, Y )〉
The second fundamental form determines the curvature of N is terms of the curvature of M.
Proposition 1.12. (The Gauss equations) For vector fields X, Y, Z tangent N we have:
RN^ (X, Y, W, Z) = RM^ (X, Y, Z, W )−〈II(X, Z), II(Y, W )〉+〈II(X, W ), II(Y, Z)〉
Proof. Suppose without loss of generality that X, Y, Z, W are vectors field on M which are tangent along N and that X and Y commute. Then we have
〈∇MX ∇MY Z, W 〉 = 〈∇MX (∇NY Z + II(Y, Z), W 〉 = 〈∇NX ∇NY Z + II(X, ∇NY Z) + ∇MX II(Y, Z), W 〉 = 〈∇NX ∇NY ZW 〉 − 〈II(Y, Z), II(X, W )〉.
and hence
RN^ (X, Y, W, Z) = RM^ (X, Y, Z, W )−〈II(X, Z), II(Y, W )〉+〈II(X, W ), II(Y, Z)〉
Thus the curvature of Sn^ is
R(X, Y, Z, W ) = −〈X, Z〉〈Y, W 〉 + 〈X, W 〉〈Y, Z〉.
So far nothing we have done really required the inner product on the tangent space to be definite merely non-degenerate. As an example lets compute the curvature of hyperbolic space. Let Hn^ be the component of the hyperboloid
c = −x^20 + x^21 +... x^2 n = − 1
containing (1, 0 , 0 ,... , 0). Consider the Rn+1^ with the Lorentz inner product.
m = −dx^20 + dx^21 +... dx^2 n.
Then 1 2
dc = −dx 0 + dx 1 +... + dxn
so the normal vector using m to Hn^ is still
nˆ = (x 0 , x 1 ,... , xn).
and so II(X, Y ) = −m(X, Y )ˆn
and so m(X, Z)m(Y, W ) − m(X, W )m(Y, Z)
exactly the opposite of the sphere.
Let R denote the sub representation of Rn^ ⊗ Rn^ ⊗ Rn^ ⊗ Rn^ consisting of
Metrics g and ˜g are said to be conformal if there is a function σ so that
˜g = e^2 σg
which can be rewritten as
R˜^4 ,^0 = e^2 σ(R^4 ,^0 + g ./ (∇dσ − dσ ⊗ dσ +^1 2
|∇σ|^2 g)
From which we glean W˜ 4 ,^0 = e^2 σW 4 ,^0
or that the (3,1)-Weyl curvature is conformally invariant. Notice then that the Weyl curvature is an obstruction to a metric being locally conformally equivalent to the standard flat metric. We can derive the behavior of the Ricci curvature Ric and the scalar curvature s under conformal change of metric. Let cg denote the Ricci con- traction for the metric g. If e 1... , en is a local orthonormal frame then
cg(R)(ej , ek) =
∑^ n
i=
R(ej , ei, ek, ei)
so that Ric = −cg(R)
We will need the following.
Lemma 1.14. For all h ∈ Sym^2 (T ∗X) we have
cg(h ./ g) = trg(h) + (n − 2)h
Proof. A tedious calculation with indices.
Lemma 1.15. If g˜ = e^2 σg then we have
Ric = Ric + (∆^ ˜ σ + (n − 2)|∇σ|^2 )g + (n − 2)(∇dσ − dσ dσ) (6)
and s ˜ = e−^2 σ(s + 2(n − 1)∆σ + (n − 1)(n − 2)|∇σ|^2 ) (7)
Let γ : (a, b) → X be a smooth path. The energy of γ is
E(γ) =
∫ (^) b
a
|γ∗(
∂t
)|^2 dt
and the length is
L(γ) =
∫ (^) b
a
|γ∗(
∂t
)|dt.
A geodesic is a path which locally minimizes the length in the following sense. A variation of γ is a function F : (−, ) × (a, b) → X so that F (0, t) = γ(t). The infinitesimal variation of γ corresponding to F is the vector field along γ S = F∗( (^) ∂s∂ ). We denote by T is the tangent vector field along γ, T = γ∗( (^) ∂t∂ ). We have the following two important formulae. The first variational formula for the energy:
Lemma 1.16.
∂ ∂s
E(γs) = −
∫ (^) b
a
〈S, ∇T T 〉dt + 〈S, T 〉|ba (8)
and the first variational formula for the length:
∂ ∂s
L(γs) = −
∫ (^) b
a
〈S, ∇T /|T |T 〉 + 〈S, T /|T |〉|ba (9)
Proof.
∂ ∂s
E(γs) =
∫ (^) b
a
〈T, ∇S T 〉dt
∫ (^) b
a
〈T, ∇T S〉dt
∫ (^) b
a
∂t
〈T, S〉 − 〈∇T T, S〉)dt
= 〈S, T 〉|ba −
∫ (^) b
a
〈∇T T, S〉dt
The proof for the length is similar and left to the reader.