21 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 := f(x; y; z) 2 R j x + y + z = R g: 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 2 S2 is explicitly kp - qk dist(p; q) = 2R sin-1 : 2R Consider a right triangle with bothp 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 p h = (p; q) = 2R -1 p1 (`=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 equi- lateral triangle on S2 with side length `, the angle is 2 -1 1 ` ≈ π + p1 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,! 1 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 1 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 correc- tions 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 approxi- mately 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 2 M of a manifold is denoted TxM and called the tangent space at x. The union of all these tangent spaces is de- noted 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 2 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 : M E such that p◦ : M M is the identity map. In other words, for each x 2 M, (x) 2 Ex. (“Smooth” means that E is also a smooth manifold and is a smooth map.)! The set of smooth! sections is denoted Γ(M; E). The product f of a function and a section is defined pointwise. We can also use an index notation for sections of a vector bundle. Writing α, technically denotes the components of 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 2 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 2 TxM is a vector at x 2 M, and let " be an infinitesimal again. A connection on E allows us to identify Ex with Ex+"v; the path between these points is irrelevant to order ". Now suppose that is a smooth section of E. Using the connection, we can compare (x) with (x + "v) and define (x + "v)- (x) rv = lim : " 0 " v; w 2 T M x x + "(v + w) x + "v If x !, then if we go from to via it is irrelevant at order ". So, rv+w = rv + rw : If we multiply by a function f 2 C (M), then f(x + "v) (x +1"v)- f(x) (x) rv(f ) = lim " 0 " [f(x + "v)- f(x)] (x + "v) + f(x)[ (x + "v)- (x)] = lim! " 0 " = v(f) (x) + f(x)rv : ! Because the identification of vector spaces is linear, this is also additive rv( + ρ) = rv + rvρ.