5 Constructions of connections

Transcription

1 [under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M is called a connection on M. Historically connections on manifolds were the the first examples of connections identified explicitly. Their theory developed in the beginning of the XXth century preceded the general theory of connections in vector bundles. In the hindsight it is clear that considering connections on manifolds in the framework of arbitrary vector bundles simplifies the theory. Following tradition, for a connection on a manifold, we use Γ j ai for the connections coefficients. More precisely, if e i is a local basis of vector fields and x a are local coordinates, then a e i = Γ j ai e j. In particular, for the coordinate frame e a = a we have a e b = Γ c abe c. The connection coefficients Γ j ai or Γc ab of a connection on a manifold are also traditionally referred to as Christoffel symbols, for the German mathematician who made a substantial input in connection theory in its early years. There is one construction peculiar for connections on manifolds. Since in the covariant derivative X Y both X and Y are now of same nature, vector fields on M, it makes sense to consider its alternation What it could be? X Y Y X. Example 5.1. Consider the natural connection on R n. In Cartesian coordinates we have X Y = (X i i Y j ) e j. Therefore X Y Y X = ( X i i Y j Y i i X j) e j = [X, Y ]. 1

2 It follows that it is natural to compare X Y Y X with the commutator of vector fields [X, Y ]. Theorem-definition. On a manifold M with a connection, the expression X Y Y X [X, Y ] =: T (X, Y ) is bilinear over C (M) and thus defines a tensor field on M of type ( 1 2), called the torsion (or the torsion tensor) of the connection. Proof. We shall give two variants of a proof, a coordinate-free version that will establish linearity over functions, and a coordinate calculation. Using only the axioms of a connection, we have T (X, fy ) = X (fy ) f Y X [X, fy ] = ( X f) Y + f X Y f Y X ( X f) Y f[x, Y ] = f X Y f Y X f[x, Y ] = ft (X, Y ), as the same holds w.r.t. the first argument X, since T (X, Y ) is skewsymmetric in X and Y. Therefore, at each point x M, T (X, Y ) is a ( bilinear function T x M T x M T x M, i.e., corresponds to a tensor of type 1 ) 2. To get an explicit expression, let us calculate in local coordinates. We obtain for T (X, Y ), T (X, Y ) = X Y Y X [X, Y ] = X a ( a Y c + Γ c aby b) e c Y a ( a X c + Γ c abx b) e c (X a a Y c Y a a X c ) e c = X a Y b Γ c abe c Y a X b Γ c abe c = X a Y b (Γ c ab Γ c ba) e c. Therefore the torsion tensor in local coordinates is T = T c abdx a dx b c = (Γ c ab Γ c ba) dx a dx b c. (1) Remark 5.1. The expression (1) is valid only in a coordinate basis of vector fields. If a different basis is chosen, the expression for the torsion tensor will be different. Definition 5.1. A connection with the zero torsion tensor is called torsionless or symmetric. 2

3 For a symmetric connection, X Y Y X = [X, Y ]. Proposition 5.1. To an arbitrary connection on a manifold uniquely corresponds a symmetric connection. Proof. Note that the difference of two connections on a vector bundle is a tensor object : X u Xu is linear in u over functions and therefore defines a linear operator in the fibers (for each fixed X). In particular, on a manifold M, the difference of two connections is a tensor field of type ( ) 1 2 and, conversely, adding to a connection any such field gives a connection, XY = X Y + S(X, Y ). It follows that there is a natural way of adding a tensor field to a connection to make it symmetric. Namely, if T is the torsion tensor of a connection, then the connection XY = X Y 1 2 T (X, Y ) will have zero torsion and this is the only choice if a skew-symmetric tensor field is added (check!). Problem 5.1. Show that the Christoffel symbols for the symmetric connection corresponding to a connection with the Christoffel symbols Γ c ab are Γ c ab = 1 2 (Γc ab + Γ c ba) For a manifold M, a metric g on the tangent bundle T M is called a metric on M. Consider a manifold M endowed with a metric g on the tangent bundle T M. A metric on M is a symmetric tensor field g of type ( 0 2) specifying a scalar product in each tangent space T x M. A manifold M together with a chosen metric g is called a Riemannian manifold. Remark 5.2. The definition of a scalar product includes the positivity condition (u, u) > 0 for any vector u 0. This in particular implies the non-degeneracy of the metric, i.e., for an arbitrary non-zero vector u it is impossible that (u, v) = 0 for all vectors v (take as v the vector u itself and use positivity to reach a contradiction). If one replaces the positivity 3

4 condition for a metric by the more general non-degeneracy condition, it will give what is called pseudo-euclidean vector spaces and for manifolds, pseudo- Riemannian manifolds. They are important, e.g., for relativity theory. The main theorem of this section holds for the pseudo-riemannian case as well. Recall that a connection on a manifold is called symmetric if it has zero torsion. Theorem 5.1 (Levi-Civita Theorem). For a Riemannian manifold there exists a unique symmetric connection compatible with metric. Proof. We have two main relations from which we wish to determine : X Y Y X = [X, Y ] (symmetry) and Z (X, Y ) = ( Z X, Y ) + (X, Z Y ) (compatibility with metric). Rewrite the second equation as ( Z X, Y ) + (X, Z Y ) Z (X, Y ) = 0 and consider cyclic permutations of the vector fields X, Y, Z: and ( Y Z, X) + (Z, Y X) Y (Z, X) = 0 ( X Y, Z) + (Y, X Z) X (Y, Z) = 0. Take the sum of these three equations with the minus sign for the first equation and the plus sign for the second and the third equation. We arrive at ( X Y, Z) + (Y, X Z) + ( Y Z, X) + (Z, Y X) ( Z X, Y ) (X, Z Y ) X (Y, Z) Y (Z, X) + Z (X, Y ) = 0. Now we can use the symmetry condition to replace X Z Z X by [X, Z] and Y Z Z Y by [Y, Z]. We can also replace Y X by X Y [X, Y ]. We obtain 2( X Y, Z) ([X, Y ], Z) + (Y, [X, Z]) + ([Y, Z], X) X (Y, Z) Y (Z, X) + Z (X, Y ) = 0 4

5 or ( X Y, Z) = 1 ( ([X, Y ], Z) ([X, Z], Y ) ([Y, Z], X) 2 ) + X (Y, Z) + Y (Z, X) Z (X, Y ). This uniquely defines an operation because of the non-degeneracy of the metric. In this way we have proved the uniqueness part, that the properties to be compatible with metric and have zero torsion uniquely define a connection on a manifold. Now we need to prove the existence part, i.e., show that such a connection exists. Since we have obtained an explicit expression for ( X Y, Z) for all X, Y and Z, we may take it as the definition and check that, firstly, it defines a connection (the axioms are satisfied); secondly, that this connection is compatible with metric and has a zero torsion. It is an easy exercise which we leave to the reader. The coordinate-free expression above can be used to deduce concrete formulas in a local frame. For example, consider local coordinates x a and the corresponding local frame e a = a. Let g ab be the metric coefficients. Since the commutators of the vector fields e a vanish, we arrive at the following classical Christoffel formula: g cp Γ p ab = 1 ) ( a g bc + b g ac c g ab 2 or Γ p ab = 1 2 gpc ( a g bc + b g ac c g ab ). Remark 5.3. Alternatively, the Christoffel formulas can be deduced directly (as it is done in classical textbooks). It is very close to the argument given above. The compatibility with metric is expressed by the formula a g bc = Γ p ab g pc + g bp Γ p ac or a g bc = Γ acb + Γ abc where we denoted Γ abc = g bp Γ p ac. The symmetry of the connection is equivalent to the condition Γ abc = Γ cba. We shall rewrite the above formula two more times applying the cyclic permutations of the indices a, b, c: c g ab = Γ cba + Γ cab, b g ca = Γ bac + Γ bca. 5

6 Now we add the three equations taking the first and the third ones with the plus sign and the second one, with the minus. We obtain a g bc c g ab + b g ca = Γ acb + Γ abc Γ cba Γ cab + Γ bac + Γ bca. At the right-hand side the terms Γ abc and Γ cba, and Γ cab and Γ bac, mutually cancel due to the connection symmetry, while the terms Γ acb and Γ bca add up to give 2Γ acb. Hence we have (taking into account also the symmetry of the metric coefficients): 2Γ acb = a g bc + b g ac c g ab. Finally, by raising the index with the help of g cp, we arrive at Γ p ab = 1 ( ) 2 gcp a g bc + b g ac c g ab, which is the desired Christoffel formula. 5.2 Holomorphic bundles and the Chern theorem A complex manifold (also complex-analytic or holomorphic) is a smooth manifold of dimension 2n with a chosen class of atlases where the 2n real local coordinates in each chart are assembled in n complex variables regarded as complex coordinates and the coordinate transformations between charts are given by complex-analytic (holomorphic) functions. (Of course, such a structure exists not for all even-dimensional manifolds). Examples include C n, the complex projective spaces CP n, and smooth algebraic varieties (manifolds specified by polynomial equations in a complex affine or projective space). We typically use z a = x a + y a as complex coordinates, where x a, y a are the corresponding real coordinates. Because the changes of coordinates on a complex manifold are holomorphic, the partial derivatives in holomorphic directions and the differentials of holomorphic coordinates transform independently from the corresponding anti-holomorphic objects: z a = z a (z ) dz a = dz a z = za a z a 6 za z a z a

7 and z a = z a (z ) ( ) z d z a a = d z a z a = ( z a z a z a ) z a. (Example for a single complex variable: suppose z = 1/z where z 0 and z 0. Then dz = 1 dz, = (z ) 2 and d z = 1 d z, (z ) 2 z z ( z ) 2 = z ( z ) 2.) z It follows that k-forms on a complex manifold can be decomposed into the sums of (p, q)-forms where p + q = k and this is invariant under changes of coordinates. (A (p, q)-form is a linear combination of exactly p holomorphic differentials dz a and exactly q anti-holomorphic differentials d z a.) For example, a 1-form ω is ω = ω a dz }{{} a + ωād z }{{} a the (1, 0)-part the (0, 1)-part The exterior differential d also decomposed into the sum of two operators, denoted and : d = + where = dz a z and = d z a a z. a The operator maps (p, q)-forms to (p+1, q)-forms and, to (p, q+1)-forms. In particular, a function f is holomorphic if f = 0. The operators and satisfy 2 = 0, 2 = 0 and + = 0 (which follows from d 2 = ( + ) 2 = 0). A vector bundle E M on a holomorphic (complex-analytic) manifold is called holomorphic is there is a bundle atlas (equivalently, a collection of local frames e α on U α where M = U α ) such that the transition functions g αβ are holomorphic functions on U α U β. The corresponding frames are called holomorphic. (A choice of such a collection of local frames is a new part of structure, which is referred to as a choice of a holomorphic structure for a given vector bundle.) Suppose for a bundle E M we have two holomorphic frames, e and e such that e = e g, where g is the transition matrix and g = g(z), i.e., 7

8 g z a = 0. Then for a section u = e i u i the operation makes perfect sense and takes sections to E-valued (0, 1)-forms: u := e i ( u i ) = e i ( u i ), because g = 0. In particular, e = 0. Hence we may consider a connection of a special kind: u = Du + u, }{{} (1,0) }{{} (0,1) where the operator D takes sections to E-valued (1, 0)-forms. We have e = De = e A, and the connection 1-form A is of type (1, 0): A = A a dz a (the components A α corresponding to z a are zero). We arrive at the following definition. Definition 5.2. A connection on a holomorphic bundle E over a complex manifold M is holomorphic if the (0, 1)-part of the covariant differential u for an arbitrary section u is just u (where u makes sense for a section due to the holomorphicity of the transition functions), i.e., = D + where D takes sections of E to E-valued (1, 0)-forms. (Note that the coefficients A a = A a (z, z) of a holomorphic connection need not be holomorphic functions and typically aren t.) Theorem 5.2 (Chern theorem). On a holomorphic vector bundle endowed with a metric there exists a unique holomorphic connection compatible with the metric. If the metric is given in a local holomorphic frame by a Hermitian matrix h = (hīj ), then the connection 1-form is given by the formula A = h 1 h. 8

9 Proof. Suppose the required connection exists. Consider a local holomorphic frame e. We know that the condition of the compatibility with a metric is expressed by the formula dh = B + B where h = (hīj ) is the Gram matrix, B = (Bīj ) where Bīj = hīk A k j, and A = (A k j) is the connection 1-form. In the matrix notation, B = ha. Since the connection is holomorphic, in a holomorphic frame the connection 1-form has type (1, 0). Hence B also has type (1, 0) and thus B has type (0, 1) (under the complex conjugation the differentials dz a become d z a ). Therefore B and B coincide with the (1, 0) and (0, 1) parts of the decomposition of the 1-form dh respectively. On the other hand these are h and h, by the definition. Hence B = h and B = h. From here we immediately arrive at A = h 1 B = h 1 h, as claimed. This proves the uniqueness part of the theorem. To show the existence of a desired connection we take the above formula as the definition of the connection 1-form, in each holomorphic frame e. One has to show that it possesses the correct transformation property under a change of frame (i.e., indeed defines a connection). Notice the following transformation law for the Gram matrix. We have hīj = (e i, e j ). Consider new basis vectors e i = e i g i i, where g = (gi i ) is the transition matrix. We obtain h = (hī j ), where hī j = (e i, e j ) = (e ig i i, e jg j j ) = gi i (e i, e j )g j j = gi i h ījg j j, i.e., h = g hg. Therefore h = (g hg). Note that g = 0, hence g = 0, because the transition matrix is holomorphic and dg = g. It follows that h = g hg + g h g and multiplying from the left by h 1 = g 1 h 1 g 1 we arrive at A = h 1 h = g 1 h 1 g 1( g hg + g h g ) = g 1 (h 1 h)g + g 1 g = g 1 Ag + g 1 g, or A = ga g 1 dgg 1 (since g = dg). The form A defined for each local holomorphic frame transforms under a change of frame exactly as a connection 1-form. Therefore the collection of all such forms defines a connection on our vector bundle. It remains to check that this connection is holomorphic and is compatible with the metric. The first follows immediately because the connection forms are (1, 0)-forms, and the second follows because by the construction they satisfy the compatibility condition dh = B+B. 9

10 Example: tautological bundle. Problem:... Find the Chern connection in a 1-dimensional holomorphic vector bundle over CP n corresponding to the Hermitian metric given by (e, e) = (1 + w w) s for a local frame e, where w = (w 1,..., w n ) are inhomogeneous coordinates. Here s R is a parameter. Last modified: 3 (16) May