4.1. Riemannian Geometry. Geometry on a Riemannian Manifold Looks Locally Approximately Like Euclidean Geometry

Home , Riemannian manifold

4. CURVATURE 4.1. Riemannian geometry. Geometry on a Riemannian manifold looks locally approximately like Euclidean geometry. Let us take as the archetypal Riemannian manifold the 2-sphere S2 of radius R. 2 3 2 2 2 2 S := {(x, y, z) ∈ R | x + y + z = R }. In the geometry on S2, the role of straight lines is played by great circles. A great circle on S2 is a circle which (in R3) is centered on the origin. Points on S2 can be given as Euclidean vectors. The distance between p, q ∈ S2 is explicitly kp − qk dist(p, q) = 2R sin−1 . 2R

Consider a right triangle with both√ legs of length `. In Euclidean geometry, the length of the hypotenuse is h = 2`, but not in spherical geometry. On S2, we can construct such a triangle with the right vertex at (R, 0, 0), and the other two vertices at p = (R cos[`/R],R sin[`/R], 0) and q = (R cos[`/R], 0, R sin[`/R]). the length of the hypotenuse is h i √ h = (p, q) = 2R −1 √1 (`/R) ≈ 2` (1 − 1 R−2`2). dist sin 2 sin 4 π In Euclidean geometry, the angles of an equilateral triangle are 3 . For an equilateral triangle on S2 with side length `, the angle is 2 −1 1 ` ≈ π + √1 R−2`2. sin 2 sec 2R 3 4 3 In Euclidean geometry, a circle of radius ` has circumference 2π`. On S2, it has circumference ` 1 −2 2 2πR sin R ≈ 2π` 1 − 6 R ` . In each of these examples, the Euclidean approximation becomes exact in the limit of either small size ` 0 or a large sphere R . Note that the leading order correction is proportional to R−2. This number is the 2 2 curvature of S . In general,→ curvature is not just a number,→ ∞ but S happens to have constant curvature. In the limit of R , the curvature goes to 0 and spherical geometry becomes Euclidean geometry. A Riemannian manifold is locally→ Euclidean ∞ to first order. This means that if we consider some geometrical figure of small size ≈ `, then its lengths and angles are approximately what they would be in Euclidean geometry. The first corrections to this approximation are of order `2 beyond the leading order. There are no corrections needed at first order in `. This is why (pseudo)Riemannian geometry is the correct mathematics for de- scribing gravity. The principle of equivalence states that there is no difference in the physics in a (relatively small) box at rest in a gravitational field and the physics in the same box being accelerated, with no gravitational field. We just have to 22 replace Euclidean geometry with Minkowskian geometry (the geometry of spe- cial relativity). The statement becomes that the geometry of spacetime is approximately Minkowskian to first order.

4.2. Parallel transport. In Euclidean space, the location of a vector is not essential. It is perfectly meaninful to say that two vectors at different points are equal, or that a vector field is constant. This is not the case on a typical Riemannian manifold. There is no consistent way of defining when vectors at different points of S2 should be considered equal. Consider a curve C on a Riemannian manifold M. Over a small (infinitesimal) neighborhood U of C, the geometry looks Euclidean. If C is not closed, then U might as well be a neighborhood of a curve in Euclidean space. If we start with a vector v at the beginning of C, then we can parallel transport v along C. If we flatten out U in Euclidean space, this means that we just keep v constant along C. Equivalently, we can move v by infinitesimal steps along C. Each step is in an infinitesimal neighborhood which looks Euclidean enough that we can make the next vector equal to the last one. This does not give a consistent way of identifying vectors at different points, because it depends upon the curve connecting two points. In fact, if we parallel transport a vector around a closed curve, then it almost never comes back to the vector we started with. This is the characteristic feature of curvature. Consider the circle C of points on S2 given by vectors which form an angle θ with the z axis. In terms of spherical geometry, this circle is the set of points at distance Rθ from the north pole. (This is not a great circle.) There is a cone (in R3) which is tangent to S2 at C. An infinitesimal neighborhood of C in S2 looks just like an infinitesimal neighborhood of C in the cone. If we cut the cone along a straight line from the vertex to C, then it can be flattened out. Now C is only part of a circle. Although C has circumference 2πR sin θ. On the flattened cone, it is part of a circle of radius R tan θ. So, the flattened cone has a wedge of angle δ = 2π(1 − cos θ) missing. If we parallel transport a vector all the way around C, then it comes back rotated by the angle δ. On the other hand, the area on S2 within the circle C is easily computed to be A = 2πR2(1 − cos θ). So, there is the simple relation δ = R−2A. In fact, this is more general; if a vector is parallel transported around a closed curve on S2, then it gets rotated by an angle proportional to the enclosed area. As was mentioned above, that proportionality R−2 is the curvature of S2. On a general Riemannian manifold, there is not such a simple formula for the parallel transport of a vector around a closed curve. The general idea is that the effect of parallel transport around a closed curve is something like the total curvature enclosed by that curve, but this is only precise for an infinitesimal closed curve. This leads to the idea of the curvature tensor, as we will see.

4.3. Connections. The space of vectors at a point x ∈ M of a manifold is denoted TxM and called the tangent space at x. The union of all these tangent spaces is denoted TM and called the tangent bundle of M. This is a manifold, but more impor- tantly, it is an example of a vector bundle over M. 23

A vector bundle over M is a topological space E with a continuous surjective map −1 p : E M and the structure of a vector space on Ex := p (x) for each x ∈ M, that is locally trivial in the sense explained below. A trivial→→ vector bundle is just the product V × M of a vector space with M, and p is the projection ignoring the V part. If U ⊆ M is an open set, then a trivialization of E over U is an identification of p−1(U) ⊆ E with a trivial vector bundle over U. “Locally trivial” simply means that E can be trivialized over some neighborhood of any point of M. Trivializations are similar to coordinate systems. In fact, a coordinate system gives a trivialization of TM over the coordinate patch. The vector space is just V = Rdim M. The generalization of a vector field is a (smooth) section of a vector bundle. A section of E is a continuous map ψ : M E such that p◦ψ : M M is the identity map. In other words, for each x ∈ M, ψ(x) ∈ Ex. (“Smooth” means that E is also a smooth manifold and ψ is a smooth map.)→ The set of smooth→ sections is denoted Γ(M, E). The product fψ of a function and a section is defined pointwise. We can also use an index notation for sections of a vector bundle. Writing ψα, technically denotes the components of ψ in local trivializations. In a trivialization, the vector spaces Ex at different points are identified, but this is not natural. For a general vector bundle, such as the tangent bundle, nothing identifies the vectors spaces at different points. However, for the tangent bundle of a Riemannian manifold, we can identify different tangent spaces in a non-canonical way. If C is a curve connecting x, y ∈ M, then parallel transport along C identifies TxM with TyM. This leads to the concept of a connection. A connection on a vector bundle E is a way of identifying the vector spaces Ex along a curve. This should respect concatenation (putting curves together, end to end). If the curve is changed a little bit, then the identification should change just a little bit. Now suppose that v ∈ TxM is a vector at x ∈ M, and let ε be an infinitesimal again. A connection on E allows us to identify Ex with Ex+εv; the path between these points is irrelevant to order ε. Now suppose that ψ is a smooth section of E. Using the connection, we can compare ψ(x) with ψ(x + εv) and define ψ(x + εv) − ψ(x) ∇vψ = lim . ε 0 ε v, w ∈ T M x x + ε(v + w) x + εv If x →, then if we go from to via it is irrelevant at order ε. So,

∇v+wψ = ∇vψ + ∇wψ. If we multiply ψ by a function f ∈ C (M), then f(x + εv) ψ(x +∞εv) − f(x) ψ(x) ∇v(fψ) = lim ε 0 ε [f(x + εv) − f(x)]ψ(x + εv) + f(x)[ψ(x + εv) − ψ(x)] = lim→ ε 0 ε = v(f) ψ(x) + f(x)∇vψ. → Because the identiﬁcation of vector spaces is linear, this is also additive ∇v(ψ + ρ) = ∇vψ + ∇vρ. 24

Doing this with a vector ﬁeld leads to the precise deﬁnition of a connection. A connection (or covariant derivative) on a smooth vector bundle E over a manifold M is a map ∇ : X(M)×Γ(M, E) Γ(M, E) such that for any f ∈ C (M), v, w ∈ X(M), and ψ, ρ ∈ Γ(M, E): ∞ → ∇v+fwψ = ∇vψ + f ∇wψ (4.1)

∇v(ψ + ρ) = ∇vψ + ∇vρ (4.2)

∇v(fψ) = v(f)ψ + f ∇vψ. (4.3) In tensor notation, we write this as α i α (∇vψ) = v ∇iψ .

Because of these properties, ∇vψ makes sense even if v is a vector at a point and ψ is only defined along a curve that v is tangent to. We can recover the geometric idea of a connection from this. A section ψ along a curve C can be considered “covariantly constant” (according to ∇) if ∇vψ = 0 along C for any vector v tangent to C. If a value for ψ is given at one point of C, then we can transport it along C by this rule. We can also transport along the flow of a vector field. Infinitesimally, this means that we construct from ψ ∈ Γ(M, E) a new section whose value at x + εv is deter- mined from ψ(x). To first order in ε, this is

ψ − ε∇vψ. To ﬂow forward ﬁnitely, just take many small steps,

t N 1 2 2 lim 1 − ∇v ψ = exp(−t∇v)ψ = ψ − t∇vψ + t ∇vψ + .... N N 2

Example.→∞A trivialization of E over U deﬁnes a “trivial” connection ∂ on E over U.A section that is constant in the trivialization is constant according to this connection. In index notation, the trivial connection for a trivialization over a coordinate patch is given by partial derivatives, α α ∂iψ := ψ,i. Remark. If ∇ and ∇0 are two connections on E, then the difference A := ∇0 − ∇ is something simpler. All derivatives have cancelled out, as we can check by 0 0 Av(fψ) = ∇v(fψ) − ∇v(fψ) = v(f)ψ + f ∇vψ − v(f)ψ − f ∇vψ = f Avψ. This means that A is just a tensor,

α i α β (Avψ) = v Aiβψ . Remark. Together, these two facts show that any connection can locally be written as a trivial connection, plus some correction. ∇ = ∂ + A

α α α β ∇iψ = ψ,i + Aiβψ Keep in mind that A here depends very much upon the choice of trivialization 25

4.4. The curvature tensor. Any connection gives a version of parallel transport along curves. Curvature is the nontriviality of transport around closed curves. So, the concept of curvature is not limited to parallel transport on the tangent bundle of a Riemannian manifold; it extends to any connection on a vector bundle,. Parallel transport around a large closed curve is generally very complicated, so in order to quantify curvature, we need to consider very small closed curves. In fact, we should consider the infinitesimal polygon (Fig. 1) formed by two vector fields u, v ∈ X(M) and their Lie bracket. Recall that on S2, the effect of parallel transport around a loop was proportional to the enclosed area. So, if we transport around this little pentagon, the effect should be approximately proportional to ε2 and we can ignore anything of order ε3 or higher. So, flow ψ by εv, then εu, ε2[u, v], −εv, and finally −εu. The result is, 2 exp(ε∇u) exp(ε∇v) exp(−ε ∇[u,v]) exp(−ε∇u) exp(−ε∇v)ψ 2 ≈ ψ + ε ∇u∇v − ∇v∇u − ∇[u,v] ψ. This suggests the definition K(u, v)ψ := ∇u∇v − ∇v∇u − ∇[u,v] ψ. (4.4) This is the curvature of the connection ∇. Even though this formula involves sev- eral derivatives of u, v, and ψ, all of these cancel. The value of K(u, v)ψ at x ∈ M only depends upon u(x), v(x) and ψ(x). To see this, check that if one of these fields is multiplied by a function f ∈ C (M), then the result is simply multiplied by f (with no derivatives of f left). ∞

K(u, fv)ψ = ∇u(f∇vψ) − f ∇v∇uψ − ∇[u,fv]ψ

= f ∇u∇vψ + u(f)∇vψ − f ∇v∇uψ − ∇u(f)v+f[u,v]ψ = f K(u, v)ψ Exercise 4.1. Check that K(u, v)(fψ) = f K(u, v)ψ. This means that the curvature is a tensor. In index notation, we can write it as: α i j α β [K(u, v)ψ] = u v K βijψ