7 Curvature of a connection

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1 [under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the general case. Let e i denote, as before, the vectors of a local basis in the tangent bundle and n stand for a unit normal vector. We have de i = e j Γ j i + nb i, dn = e i b i, where b i = g ij b j. Let us apply the exterior differential to both sides taking into account that d 2 = 0. We obtain from the first equation and 0 = de j Γ j i + e j dγ j i + dn b i + n db i 0 = de i b i e i db i from the second equation. After using the derivation equations again to express the differentials of the basis vectors we obtain and 0 = (e k Γ k j + nb j ) Γ j i + e k dγ k i e k b k b i + n db i. 0 = (e j Γ j i + nb i) b i e j db j (we have renamed the index in the last term). The normal and tangent parts should vanish separately, so by collecting the tangent terms we arrive at the equations dγ i k + Γ i j Γ j k bi b k = 0 (1) (we have swapped the indices i and k) and db j + Γ j i bi = 0, (2) 1

2 and by collecting the normal terms we arrive at and db i + b j Γ j i = 0 (3) b i b i = 0. (4) We obtained a set of relations, from which (1), (2) and (3) involve derivatives, while (4) is algebraic. Let us have a closer look at them. First of all, one can see that the algebraic relation (4) is satisfied identically due to the definition of the 1-forms b i and b i. Indeed, b i = e p b pi and b i = g ij b j = g ij e q b qj ; hence b i b i = e p b pi g ij e q b qj = e p e q b pi g ij b qj. Here we expanded b i and b i over the basis of 1-forms (e i ) dual to the basis of vector fields (e i ). Now we may notice that the coefficients of the exterior products e p e q are symmetric in p and q, because b qi g ij b pj = b qj g ji b pi = b qj g ij b pi = b pi g ij b qj (we have renamed the summation indices and used the symmetry of the metric). Therefore the sum b i b i = e p e q b pi g ij b qj vanish identically. Secondly, one can see the relations (2) and (3), which look very similar, are in fact equivalent. (Expand b i as g ik b k and substitute into (3). After taking into account the equality dg ik = Γ p i g pk +g ip Γ p k expressing the compatibility of the connection with the metric, one will arrive at an equation that is directly equivalent to (2). We leave details to the reader.) We summarize the outcome in the following theorem. Theorem 7.1. The coefficients of the second fundamental form (or the shape operator) of a hypersurface M n R n+1 are related with the first fundamental form and the connection by the identities The second identity can be equivalently written as dγ i j + Γ i k Γ k j = b i b j (5) db i + Γ i j b j = 0. (6) db i + b j Γ j i = 0. (7) The identity (5) is known as the Gauß relation and the identity (6) or (7), as the Peterson Codazzi Mainardi relation. If the tensors g ij and b ij are given, these relations are the necessary conditions that g ij and b ij represent the first and the second fundamental forms of a hypersurface, respectively. 2

3 Remark 7.1. What can be said for surfaces M n R n+k where k > 1? Curves in R 3 serve as a (degenerate) example, with n = 1. For k > 1, a basis in the normal bundle consists of more than one vector. Analogs of the derivation equations for such surfaces include connection coefficients for the normal bundle. One can also deduce the corresponding analogs of the Gauß Peterson Codazzi Mainardi relations, which will be more complicated. The simplification in the case of hypersurfaces is due to the fact that the normal bundle is one-dimensional, hence it has a natural trivial connection (the derivatives of a single unit normal vector have to be tangent to the surface). Let us concentrate on the Gauß relation (5). Introduce the object Ω i j := dγ i j + Γ i k Γ k j (8) and consider Ω ij := g il Ω l j. The Gauß relation can be re-written as Notice, in particular, that Ω i j = b i b j or Ω ij = b i b j. (9) Ω ij = Ω ji. (10) Important conclusions can be deduced from further analysis of (9). The LHS of either of the equations (9) depends only on the metric on M. Indeed, working in a coordinate basis, one can express the connection coefficients Γ i jk in terms of the components of the metric and their first derivatives; thus Ω i j or Ω ij are functions of the components of the metric and their derivatives up to the second order. Therefore, the combinations b i b j or b i b j in the RHS, a priori defined in terms of the embedding into R n+1, turn out to be depending only on the induced metric. On the other hand, the 1-forms b i and b i in the RHS are but the condensed components of the tensor fields B = b ij e i e j (the second fundamental form) and W = b i j e i e j (the shape operator) respectively. Therefore their exterior products b i b j and b i b j transform by a tensor law under a change of basis. The Gauß relation implies that the same holds for Ω i j and Ω ij, which is not obvious from the definition (8). Consider now n = 2, i.e., surfaces in three-dimensional space. The Gauß relation implies the following remarkable statement. Theorem 7.2 (Gauß s Theorema Egregium). The Gaussian curvature K of a surface M 2 R 3 depends only on the first fundamental form (the metric) of M 2 and not on the embedding in R 3. 3

4 Proof. For n = 2 we can expand the RHS of the second equation in (9) as follows: b i b j = b ki e k b lj e l = e 1 e 2 (b 1i b 2j b 2i b 1j ). This vanishes for i = j, and for i j we obtain b 1 b 2 = b 2 b 1 = det B e 1 e 2 where B = (b ij ) is the matrix of the second fundamental form. Since det B = Kg, where K is the Gaussian curvature and g = det(g ij ), the Gauß relation (9) takes the form ( ) ( ) Ω11 Ω 12 0 Kg = e 1 e 2. Ω 21 Ω 22 Kg 0 This implies the statement; the 2-form Ω 12 and therefore the Gaussian curvature can be expressed via the metric coefficients and their derivatives up to the second order. The Theorema Egregium is often stated as follows: Gaussian curvature is an isometry invariant or a bending invariant. Isometry means a map that preserves the metric, bending means deformation of an embedding into Euclidean space without changing the induced metric. Example 7.1. One can show directly that the Gaussian curvatures for a cylinder and a cone are equal to zero. This is in perfect correspondence with the fact that these surfaces can be developed on a plane (a flat surface). Example 7.2. For comparison, the mean curvature of a round cylinder of radius R equals 1/R while the mean curvature of a plane is zero. Therefore mean curvature is not an isometry invariant. Example 7.3. The Gaussian curvature of a sphere of radius R equals R 2. Therefore it follows from Theorema Egregium that it is not possible to map a piece of the sphere onto a flat surface without distorting distances and angles. This explains why it is impossible to have a perfect geographic map, i.e., such that measurements on the map give distances on the corresponding region of the Earth. We may summarize the discussion of curvature for surfaces as follows. The original notion of curvature comes from curves. The curvature of a curve in 4

5 Euclidean space measures how it is curved (different from a straight line), more precisely, the rate of rotation of the tangent. It is a function of the embedding into R n and, clearly, by bending the curve its curvature can be made zero. If a curve remains on a surface, there is a lower bound for the magnitude of the curvature depending on a direction in the tangent plane. Thus we come to the second fundamental form, which introduces a curvature of a surface by measuring how curves on it are necessarily curved. This form (together with the metric) gives numerical invariants of the curvature of a surface such as mean curvature and Gaussian curvature. Now we see that a part of curvature (such as mean curvature) depends on embedding and may change if we bend the surface preserving all lengths and angles. At the same time, some other part, namely, Gaussian curvature, is an internal curvature of a surface, meaning that it is a property of its internal geometry (lengths and angles measured on the surface) and not a particular embedding into Euclidean space. This is a completely new idea that allows a far-reaching generalization. 7.2 Curvature 2-form for vector bundles Consider a vector bundle E B with a connection. We use the object Ω i j appearing in the LHS of the Gauß relation (5) or (9) as a model for the following definition. Let A be the local connection 1-form w.r.t. a local frame e = (e i ). Recall that A is a matrix-valued 1-form. Definition 7.1. The curvature 2-form F w.r.t. a frame e is defined by the formula F = da + A 2. (11) Here the square A 2 is the matrix square. Note that the matrix entries are 1-forms and the exterior product is taken. Remark 7.2. It is possible to rewrite the definition as F = da + A 2 = da + 1 [A, A]. 2 The commutator for matrices with entries taking values in differential forms is defined by [A, B] = AB ( 1) pq BA 5

6 where the entries of A are p-forms and the entries of B are q-forms. Hence [A, A] = 2A 2. This way of expressing curvature has the advantage of doing it entirely in terms of the Lie algebra structure. For the Levi-Civita connection on surfaces in Euclidean space, Definition 7.1 gives the internal curvature of the surface. As follows from the Gauß relation, the internal curvature is a tensor object (because the shape operator is a tensor object). Let us show that the same holds in general. Suppose we have two local frames e α and e β so that e α = Φ αβ e β, where Φ αβ is the transition matrix (α and β are indices denoting domains where the local frames are defined, and the transition holds on the intersection). Denote by A α, F α, etc., the corresponding connection and curvature forms. Theorem 7.3. The curvature 2-forms transform according to the following law: F α = Φ αβ F β Φ 1 αβ. Proof. We know the transformation law for the connection forms: Hence A α = Φ αβ A β Φ 1 αβ dφ αβ Φ 1 αβ. F α = da α +A 2 α = d ( Φ αβ A β Φ 1 αβ dφ ) ( αβ Φ 1 αβ + Φαβ A β Φ 1 αβ dφ αβ Φαβ) 1 2 = dφ αβ A β Φ 1 αβ + Φ αβda β Φ 1 αβ + Φ αβa β Φ 1 αβ dφ αβφ 1 αβ dφ αβ Φ 1 αβ dφ αβφ 1 αβ + Φ αβ A 2 βφ 1 αβ Φ αβa β Φ 1 αβ dφ αβ Φ 1 Φ αβ da β Φ 1 αβ + Φ αβa 2 βφ 1 αβ = Φ αβ αβ dφ αβ A β Φ 1 αβ + dφ αβ Φ 1 αβ dφ αβ Φ 1 αβ ( = daβ + Aβ) 2 Φ 1 αβ = Φ αβf β Φ 1 αβ. Corollary 7.1. The matrix-valued local 2-forms F α represent a global 2-form F on B with values in operators acting on the fibers of E. Proof. Indeed, under changes of bases F α transform precisely as matrices of a linear operator. Each component of the 2-form F α at a point x B can be regarded as the matrix of a linear operator acting on the vector space E x w.r.t. the basis e α (x), and the whole form F α, as the matrix of an operatorvalued form. 6

7 We use boldface for the curvature 2-form when considered as an operatorvalued form on B rather than a collection of local matrix-valued forms: F Ω 2 (B, End E). Remark 7.3. Unlike the curvature 2-form, the connection 1-form makes sense only as a collection of local matrix-valued forms and cannot be interpreted as an operator-valued form. Example 7.4. For a line bundle (i.e., a vector bundle with one-dimensional fibers) E B, operators on fibers are simply numbers, therefore the curvature 2-form is a well-defined ordinary 2-form F on B. By the definition, locally F = F α = da α, because the square A 2 α in this case is zero. We have F α F β = 0 on intersections. It follows that here F is automatically closed (but not necessarily exact, since A α exist only locally, in general). In the general case the curvature form (as a matrix-valued local 2-form F defined w.r.t. a local frame) is not closed. Closedness is replaced by the following property. Theorem 7.4 (Bianchi identity). The local curvature 2-form F satisfies the identity df + [A, F ] = 0 (12) where A is the connection 1-form. Proof. Directly: df = d ( da + A 2) = da A A da = (F A 2 ) A A (F A 2 ) = F A A 3 A F + A 3 = [A, F ]. Remark 7.4. The definition of the curvature form (11) and the Bianchi identity (12) are sometimes written together as da = A 2 + F, (13) df = [A, F ] (14) 7

8 and called the structure equations. For historical reasons equation (13) is also referred to as the second structure equation and the Bianchi identity (12), (14), as the second Bianchi identity. (The so-called first structure equation and first Bianchi identity make sense only for connections on manifolds. We do not discuss them here.) Note that discussing the Bianchi identity we were careful to speak in terms of local matrix-valued forms and not the End E-valued form F. Applying d does not make sense for F. There still is a way of interpreting the Bianchi identity in terms of F, see below. We first need to discuss further properties of curvature. What is the geometric meaning of curvature of a connection? We have arrived at Definition 7.1 by extending a notion for surfaces. There are several other ways leading to the notion of curvature for vector bundles and in particular explaining the formula (11). We can arrive at this notion by considering either of the following (not exhaustive list): Commutators of covariant derivatives; The square of the exterior covariant differential (to be defined); Obstructions to making connection trivial. We know that a b b a = 0. What about a b b a? It makes sense to consider covariant derivatives along arbitrary vector fields, comparing the commutator [ X, Y ] with the covariant derivative along [X, Y ]. Theorem 7.5. For arbitrary vector fields X, Y X(B) and a section u Γ(B, E), the expression [ X, Y ]u [X,Y ] u is linear over functions in u, X and Y. Therefore it defines a bilinear function on T x B T x B with values in linear operators on E x. Proof. Suppose Y is replaced by fy. Then [ X, fy ] = [ X, f Y ] = ( X f) Y + f[ X, Y ] and [X,fY ] = ( X f)y +f[x,y ] = ( X f)y + f[x,y ] = ( X f) Y + f [X,Y ]. Hence F (X, fy )u = [ X, fy ]u [X,fY ] u = f ( [ X, Y ]u [X,Y ] u ) = ff (X, Y )u. Suppose u is replaced by fu. Then... 8

9 Curvature as d Consider now the special case of the curvature of a connection compatible with metric. For simplicity, consider the real case first. We know that in an orthonormal frame the connection 1-form of a metric connection A α takes values in antisymmetric matrices. Therefore the matrix-valued curvature 2-form w.r.t. an orthonormal frame F α = da α [A α, A α ] also takes values in antisymmetric matrices, because differentiation does not change symmetry and the space of antisymmetric matrices is closed under commutator. (It is the Lie algebra of the orthogonal group.) Now we can notice that this has a meaning independent of any frame: the operator-valued 2-form F takes values in antisymmetric linear operators on the fibers of our vector bundle (which are Euclidean spaces). In particular, if we lower indices for F using the metric, the resulting 2-form will take values in antisymmetric bilinear forms and hence the corresponding matrices will always be antisymmetric, in any local frame, orthonormal or not. We have arrived at the following statement. Proposition 7.1. For a metric connection on a real Euclidean vector bundle, the curvature 2-form F considered as an operator-valued form takes values in antisymmetric operators. Its matrix representation takes values in antisymmetric matrices if the frame is orthonormal. The matrix representation with lower indices is always antisymmetric (in any frame). To avoid any possible confusion, let us note that the curvature 2-form is represented, in the maximal detail, by components with four indices: F i abj where a, b correspond to a frame on the base (e.g., given by coordinates) and i, j are internal indices, i.e., corresponding to a frame in the fiber. Since we have an exterior 2-form, there is always antisymmetry in a, b: F i abj = F i baj. On the other hand, antisymmetry in i, j is a special feature of a metric connection and it holds for components with lowered indices: F abij = F abji where F abij = g ik F k abj. 9

10 There is an analogous statement for the complex case, where anti-hermitian operators replace antisymmetric operators. Proposition 7.2. For a metric connection on a complex Hermitian vector bundle, the curvature 2-form F considered as an operator-valued form takes values in anti-hermitian operators. Its matrix representation takes values in anti-hermitian matrices in an orthonormal frame. The matrix representation with lower indices is anti-hermitian in any frame: F abi j = F abjī where F abi j = h k j F k abi. Example 7.5. Suppose we have a holomorphic Hermitian bundle over a complex manifold. Then, by the Chern theorem, there is a unique metric connection compatible with the complex structure. Its local connection 1-form is given in a holomorphic frame by the formula A = h h 1. Here h = (h i j ) is the Gram matrix of the frame and h 1 = (hīj ). Note that in general a holomorphic frame is not orthonormal; if it is, then A is zero. Hence A is not anti-hermitian, if not zero. However, the curvature 2-form will take values in anti-hermitian operators. Let us analyze the particular case of line bundles (dimension of fiber is one). A Hermitian metric is given by a single real function h. We have A = ln h and therefore since d = + and 2 = 0. Hence i.e., F is purely imaginary. F = d( ln h) = ln h, F = ln h = ln h = F, 7.3 The Riemann Christoffel tensor Consider a connection on a manifold M. Let Γ = Γ a dx a = (Γ i ajdx a ) be a local connection 1-form w.r.t. a local basis of vector fields e i (not necessarily related with coordinates). We consider the corresponding curvature 2-form (as a matrix-valued form) Ω = dγ + Γ 2. 10

11 In components, Ω i j = 1 ( ) 2 dxa dx b a Γ i bj bγ i aj + Γ i ak Γk bj Γi bk Γk aj. Definition 7.2. The tensor field where R = R i abj dxa dx b e i e j R i abj = aγ i bj bγ i aj + Γ i ak Γk bj Γi bk Γk aj is called the Riemann, or Riemann Christoffel, tensor. Theorem 7.6. The Riemann Christoffel tensor has the following symmetries: 1. For an arbitrary connection, 2. For a symmetric connection, 3. For a connection compatible with metric, R c abd = Rc bad. (15) R c abd + Rc dab + Rc bda = 0. (16) R abcd = R abdc. (17) Corollary 7.2. For a symmetric connection compatible with metric (Levi-Civita connection), R abcd = R cdab. (18) Last modified: 1 (14) May