Riemannian Geometry
1 Riemannian Metric
Definition 1. A Riemannian metric on a manifold M is a (0, 2)-tensor field g on M that is symmetric and positive-definite. 1 n If (E1,...,En) is a local frame of TM, and (dx , . . . , dx ) is its dual coframe, then we can write the Riemannian metric in terms of its components i j g = gijdx ⊗ dx , (1) A Riemannian metric determines an inner product on each tangent space: hX,Y i = g(X,Y ). Proposition 1. Every smooth manifold M has a Riemannian metric. Given two Riemannian manifolds (M, g) and (M 0, g0), a diffeomorphism ϕ: M → M 0 is called an isometry if ϕ∗g0 = g. If M = M 0 and g = g0, then ϕ is said to be an isometry of M.
1.1 Lengths and Distances Given a curve γ : I → M on a Riemannian manifold, we can use the metric to define the length of the curve: Z b L(γ) = hγ,˙ γ˙ i1/2dt a Note that this is independent of parametrization. Using admissible curves, we can define the Riemannian distance d(p, q) between two points p, q ∈ M to be the infimum of the lengths of all the admissible curves connecting the two points. One can show that the manifold along with the Riemannian distance is a metric space.
1.2 Pseudo-Riemannian Metric Definition 2. A pseudo-Rimenanian metric is a 2-tensor field that is symmetric and non- degenerate. Since Riemannian metrics are positive-definite, all Riemannian metrics are pseudo-Riemannian metrics. At any point p ∈ M, a pseudo-Riemannian metric can be written in the form of g = −(dx1)2 − · · · − (dxk)2 + (dxk+1)2 + ··· + (dxn)2 by a change of basis. The maximum dimension of a subspace on which g is negative definite k is called the index of g, which is invariant under a change of basis due to Sylvester’s law of inertia.
1 2 Connections
In the Euclidean plane, we usually say that a line between two points is straight if it minimizes the length of curves between the two points. However, an easier way to generalize this notion to manifolds is to view straight lines as curves with no acceleration. The problem is that given a curve γ on a manifold, we do not have a way to measure how fast its velocity vectors V (t) change since we cannot subtract vectors in different tangent spaces. In order to measure the acceleration of a curve on a manifold, we introduce the concept of the covariant derivative.
Definition 3. Let E be a vector bundle over M with Γ(E) the space of smooth sections of E. A connection is a map ∇: X(M)×Γ(E) → Γ(E), denoted by (X,Y ) 7→ ∇X Y , such that: • ∇ is C∞(M)-linear over T(M)
• ∇ is R-linear over E(M)
• ∇ satisfies the product rule ∇X (fY ) = f∇X Y + (Xf)Y If E(M) = X(M), then ∇ is called an affine connection.
∇X Y is called the covariant derivative of Y in the direction of X. Note that ∇X Y |p only depends on the value of X at p and the values of Y in some neighborhood of p. From now on, all connections will be assumed to be affine connections. In a local coordinate frames with {∂i} as the basis of the tangent spaces, we can define the Christoffel symbols of ∇ with respect to this frame by
k ∇∂i (∂j) = ΓijEk. (2)
i i Then given X = X ∂i and Y = Y ∂i, we can explicitly write ∇X Y in terms of its components:
k i j k ∇X Y = (XY + X Y Γij)∂k (3)
Example 1. On Rn, the Euclidean connection is defined by
j ∇X Y = (XY )∂j (4)
Proposition 2. Every manifold has an affine connection.
An affine connection only acts on vector fields, but any affine connection also induces connections on all tensor fields on M.
Theorem 3 (Covariant Derivatives of Tensor Fields). Let ∇ be an affine connection on M. Then ∇ induces a unique connection, also denoted ∇, on each tensor field such that
∞ • ∇ agrees with the original affine connection on X(M), and ∇X f = X(f) on C (M)
• ∇ commutes with any contraction: C(∇X Y ) = ∇X (CY )
• ∇ satisfies the product rule ∇X (Y ⊗ Z) = Y ⊗ (∇X Z) + (∇X Y ) ⊗ Z
2 k k We can use the covariant derivative of a -tensor field T to define a -tensor l l + 1 i 1 field. Let Yi ∈ X(M) and ω ∈ Ω (M), then the total covariant derivative of a tensor field T is 1 k 1 k ∇T (ω , . . . , ω ,Y1,...,Yl,X) = ∇X T (ω , . . . , ω ,Y1,...,Yl). (5) One can think of the covariant derivative of a tensor field as directional derivatives while the total covariant derivative is the total derivative of the tensor field.
2.1 Covariant Derivatives Along Curves Now we would like to define what the directional derivatives of a vector field along a curve means. Definition 4. A vector field along a curve γ is a smooth map V : I → TM such that V (t) ∈ Tγ(t)M. We denote the space of such vector fields along γ by T(γ). Note that this is not simply restricting a vector field to a curve since a self-intersecting curve may have different vectors at the intersection. However, vector field V along γ is said ˜ ˜ to be extendible if there is a vector field V such that V (t) = Vγ(t).
Theorem 4. Let ∇ be an affine connection on M. ∇ induces a unique operator Dt : T(γ) → T(γ) such that for each curve γ : I → M, Dt satisfies: • R-Linearity ˙ ∞ • Product rule: Dt(fV ) = fV + fDtV, f ∈ C (I)
0 0 • If V is extendible, then for any extension V of V , DtV (t) = ∇γ˙ (t)V where V ∈ T(γ). DtV is called the covariant derivative of V along γ.
In local coordinates, DtV can be written as ˙ k j i k DtV (t) = V (t) + V (t)γ ˙ (t)Γij(γ(t)) ∂k. (6)
2.2 Parallel Transport Definition 5. A vector field V along a curve γ is parallel along γ with respect to some affine connection ∇ if DtV = 0.
Theorem 5 (Parallel Transport). For any curve γ : I → M and vector V0 ∈ Tγ(t0)M, there exists a unique parallel vector field V along γ such that V (t0) = V0. V (t) is called the parallel transport of V0 along γ. Proof. Suppose γ(I) is contained in a single coordinate chart. Then, by equation (??), V is parallel along γ if and only if ˙ k j i k V (t) = −V (t)γ ˙ (t)Γij(γ(t)). (7) Existence and uniqueness for linear ODEs guarantees a unique V for any initial condition V (t0) = V0. If γ(I) spans multiple charts, then a unique solution exists in each local chart. By uniqueness, the solutions agree on their intersections.
3 2.3 Riemannian Connections Definition 6. Let g be the Riemannian metric of M, an affine connection ∇ is said to be compatible with g if ∇X hY,Zi = h∇X Y,Zi + hY, ∇X Zi (8) Theorem 6. For a connection ∇ on a Riemannian manifold with metric g, the following conditions are equivalent:
• ∇ is compatible with g
• ∇g = 0
• if X,Y are vector fields along γ, then d hX,Y i = hD X,Y i + hX,D Y i dt t t
2 Definition 7. The torsion tensor of an affine connection ∇ is a -tensor field T : X(M)× 1 X(M) → X(M) defined by
T (X,Y ) = ∇X Y − ∇Y X − [X,Y ]
An affine connection is said to be symmetric if it is torsion-free. When the Christoffel symbol k k is written with respect to a coordinate frame, symmetry of ∇ is equivalent to Γij = Γji. Theorem 7 (Fundamental Theorem of Riemannian Geometry). Let M be a Rie- mannian or pseudo-Riemannian manifold with the metric g. There is a unique connection on M that is symmetric and compatible with g. We call this the Levi-Civita connection, or the Riemannian connection of g.
Proof. We prove uniqueness of ∇ first. Suppose such a ∇ exists, then compatibility implies that for vector fields X,Y,Z,
XhY,Zi = h∇X Y,Zi + hY, ∇X Zi
Y hZ,Xi = h∇Y Z,Xi + hZ, ∇Y Xi
ZhX,Y i = h∇Z X,Y i + hX, ∇Z Y i
Using the symmetry condition, these three equations become
XhY,Zi = h∇X Y,Zi + hY, ∇Z Xi + hY, [X,Z]i
Y hZ,Xi = h∇Y Z,Xi + hZ, ∇X Y i + hZ, [Y,X]i
ZhX,Y i = h∇Z X,Y i + hX, ∇Y Zi + hX, [Z,Y ]i
4 These three equations gives
XhY,Zi − Y hZ,Xi + ZhX,Y i =
2h∇X Y,Zi + hY, [X,Z]i + hX, [Z,Y ]i − hZ, [Y,X]i
1 =⇒ h∇ Y,Zi = (XhY,Zi − Y hZ,Xi + ZhX,Y i X 2 (9) − hY, [X,Z]i − hX, [Z,Y ]i + hZ, [Y,X]i) If ∇1 and ∇2 both satisfies the conditions for a Levi-Civita connection, then both of them satisfies equation (9). However, the right hand side does not depend on the choice of con- 1 2 1 2 nection, so h∇X Y − ∇X Y,Zi = 0 for any X,Y,Z. This implies that ∇ = ∇ . Now we need to show that such a connection does indeed exist. For a coordinate chart (U, ϕ) of M,[∂i, ∂k] = 0, so we can write (9) as 1 h∇ ∂ , ∂ i = (∂ h∂ , ∂ i − ∂ h∂ , ∂ i + ∂ h∂ , ∂ i) ∂i j k 2 i j k j k i k i j 1 =⇒ Γmg = (∂ g + ∂ g i + ∂ g j) (10) ij ml 2 i jk j k k i Multiplying by the inverse matrix glk, we get 1 Γk = glk(∂ g + ∂ g + ∂ g ) (11) ij 2 i jk j ki k ij This defines a symmetric connection on U. The connection is also compatible with g, since ∇g = 0 as can be easily checked. This shows that a Riemannian connection exists in all coordinate charts of M. Since the connection is unique, the connections agree on the overlaps of the coordinate charts, so a unique Riemannian connection exists on M. Equation (11) allows us to compute the Christoffel symbols of a Riemannian connection in terms of gij.
2.4 Geodesics We want to define geodesics as curves whose accelerations are zero. This is equivalent to saying that geodesics are curves that parallel transport their velocities.
Definition 8. A geodesic with respect to ∇ is a curve γ such that Dtγ˙ = 0. Geodesics whose lengths equal the Riemannian distance between its two endpoints are said to be minimizing geodesics. If the connection is a Riemannian connection, the geodesic is said to be a Riemannian geodesic.
Theorem 8. Let M be a manifold with a connection. For any p ∈ M and V ∈ TpM, there exists a geodesic γ : I → M such that γ(t0) = p, γ˙ (t0) = V . Any two such geodesics agree on their common domain.
5 Proof. By definition, a curve γ(t) = (x1(t), . . . , xn(t)) is a geodesic if and only if it satisfies
k i j k x¨ (t) = −x˙ (t)x ˙ (t)Γij(x(t)) (12)
Equation (12) is called the geodesic equation. We can set vi =x ˙ i to turn the geodesic equation into a system of first order ODEs.
x˙ k(t) = V k(t)
˙ k i j k V (t) = −V (t)V (t)Γij(x(t)) Then by the existence and uniqueness theorem for first-order ODEs, there exists a unique γ in some neighborhood of t0 that satisfies the initial conditions.
A consequence of this theorem is that for any p ∈ M and V ∈ TpM, there is a unique maximal geodesic γV that cannot be extended to any larger interval. If all the maximal geodesics on a Riemannian manifold are defined for all t ∈ R, the the manifold is said to be geodesically complete. The following theorem connects this with the earlier idea of Rimennian manifolds as metric spaces:
Theorem 9 (Hopf-Rinow Theorem). A connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space.
Definition 9. Let M be a Riemanian manifold. The curvature endomorphism of M is the map T(M) × T(M) × T(M) → T(M) defined by
R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z (13)
n k k On R , using the Euclidean connection, we get that ∇X ∇Y Z = ∇X (YZ ∂k) = XYZ ∂k k k and ∇Y ∇X Z −∇Y (XZ ∂k) = YXZ ∂k. Thus, a manifold is said to be flat if R(X,Y )Z = 0. We can express R in local coordinates:
l i j k R = Rijkdx ⊗ dx ⊗ dx ⊗ ∂l, (14) where the coefficients are given by
l R(∂i, ∂j)∂k = Rijk∂l. (15)
Using the curvature endomorphism, we can define the covariant curvature tensor Rm by
Rm(X,Y,Z,W ) = hR(X,Y )Z,W i (16)
In coordinates, the curvature tensor looks like
l i j k l i j k l Rm = glmRijkdx ⊗ dx ⊗ dx ⊗ dx ≡ Rijkldx ⊗ dx ⊗ dx ⊗ dx (17)
6 3.1 Ricci and Scalar Curvatures We can summarize the information contained in a curvature tensor by tracing over some of its components. The Ricci curvature and the scale curvature are two important tensors constructed this way.
Definition 10. The Ricci curvature, denoted Rc, is a 2-tensor field whose components are defined by k km Rij = Rkij = g Rkijm (18) The scalar curvature S is defined as the trace of the Ricci tensor:
ij S = g Rij (19)
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