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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 ': M ! M 0 is called an isometry if '∗g0 = g. If M = M 0 and g = g0, then ' 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 2 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 2 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 r: X(M)×Γ(E) ! Γ(E), denoted by (X; Y ) 7! rX Y , such that: • r is C1(M)-linear over T(M) • r is R-linear over E(M) • r satisfies the product rule rX (fY ) = frX Y + (Xf)Y If E(M) = X(M), then r is called an affine connection. rX Y is called the covariant derivative of Y in the direction of X. Note that rX Y jp 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 f@ig as the basis of the tangent spaces, we can define the Christoffel symbols of r with respect to this frame by k r@i (@j) = ΓijEk: (2) i i Then given X = X @i and Y = Y @i, we can explicitly write rX Y in terms of its components: k i j k rX Y = (XY + X Y Γij)@k (3) Example 1. On Rn, the Euclidean connection is defined by j rX 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 r be an affine connection on M. Then r induces a unique connection, also denoted r, on each tensor field such that 1 • r agrees with the original affine connection on X(M), and rX f = X(f) on C (M) • r commutes with any contraction: C(rX Y ) = rX (CY ) • r satisfies the product rule rX (Y ⊗ Z) = Y ⊗ (rX Z) + (rX 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 2 X(M) and ! 2 Ω (M), then the total covariant derivative of a tensor field T is 1 k 1 k rT (! ;:::;! ;Y1;:::;Yl;X) = rX T (! ;:::;! ;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) 2 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 r be an affine connection on M. r induces a unique operator Dt : T(γ) ! T(γ) such that for each curve γ : I ! M, Dt satisfies: • R-Linearity _ 1 • Product rule: Dt(fV ) = fV + fDtV; f 2 C (I) 0 0 • If V is extendible, then for any extension V of V , DtV (t) = rγ_ (t)V where V 2 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 r if DtV = 0. Theorem 5 (Parallel Transport). For any curve γ : I ! M and vector V0 2 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 r is said to be compatible with g if rX hY; Zi = hrX Y; Zi + hY; rX Zi (8) Theorem 6. For a connection r on a Riemannian manifold with metric g, the following conditions are equivalent: • r is compatible with g • rg = 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 r is a -tensor field T : X(M)× 1 X(M) ! X(M) defined by T (X; Y ) = rX Y − rY 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 r 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 r first. Suppose such a r exists, then compatibility implies that for vector fields X; Y; Z, XhY; Zi = hrX Y; Zi + hY; rX Zi Y hZ; Xi = hrY Z; Xi + hZ; rY Xi ZhX; Y i = hrZ X; Y i + hX; rZ Y i Using the symmetry condition, these three equations become XhY; Zi = hrX Y; Zi + hY; rZ Xi + hY; [X; Z]i Y hZ; Xi = hrY Z; Xi + hZ; rX Y i + hZ; [Y; X]i ZhX; Y i = hrZ X; Y i + hX; rY Zi + hX; [Z; Y ]i 4 These three equations gives XhY; Zi − Y hZ; Xi + ZhX; Y i = 2hrX Y; Zi + hY; [X; Z]i + hX; [Z; Y ]i − hZ; [Y; X]i 1 =) hr 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 r1 and r2 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 hrX Y − rX Y; Zi = 0 for any X; Y; Z. This implies that r = r .
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