Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July)
Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗ . vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { } ∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. Connections on Vector Bundles = . + . . (3) under a change of the local frame field, which is the A connection on a fiber bundle [7] is a device that defines transformation formula for a connection, a most important a notion of parallel transport on the bundle, that is, a way formula in differential geometry. to connect or identify fibers over nearby points. If the We may assume that { } is locally finite, and { } is a fiber bundle is a vector bundle, then the notion of parallel corresponding sub-ordinate partition of unity such that transport is required to be linear. Such a connection is supp ⊂ . When ∩ ≠ ∅, there naturally exists a equivalently specified by a covariant derivative, which is non-degenerate matrix of smooth functions on ∩ an operator that can differentiate sections of that bundle such that along tangent directions in the base manifold [3]. = . , ≠ 0 (4)
Connections in this sense generalize, to arbitrary vector For every ∈ , choose an arbitrary × matrix of bundles, the concept of a linear connection on the tangent differential 1-forms on . Let bundle of a smooth manifold, and are sometimes known = . . as linear connections. Nonlinear connections are ∈ connections that are not necessarily linear in this sense. + . . (5) Definition 1. A connection on a vector bundle is a map
where the terms in the sums over with ∩ = ∅ are ∶ Γ( ) → zero. Then is a matrix of differential 1-forms on . We Γ( ∗ ⊗ ) (1) need only demonstrate the following transformation formula which satisfies the following conditions: for ∩ ≠ ∅: (i) For any , ∈ Γ( ), = . + . . . (6)
( + ) = + This can be done by a direct calculation. First observe that (ii) For ∈ Γ ( ) and any ( ) , when ∩ ∩ ≠ ∅, the following is true in the ( ) = ⊗ + intersection:
Suppose is a smooth tangent vector fields on . = . and ∈ Γ ( ). Let Thus on ∩ ≠ ∅ we have
192 Md. Showkat Ali et.al
Thus is the desired local frame field. □ . . = . . . Theorem 3. Suppose , are two arbitrary smooth tangent ∩ ∩ ∅ vector fields on the manifold . Then + . . . ( , ) = − − [ , ] (9) = − . Proof. Because the absolute differential quotient and the This is precisely (6). We see from the above that there is curvature operator are local operators, we need only consider much freedom in the choice of a connection. This the operations of both sides of (9) on a local section. completes the proof of the theorem. □ Suppose ∈ Γ( ) has the local expression Remark 1. In particular, if we let = 0 in (6), then = we obtain a connection on whose connection matrix on is Then = . . = ∑ ( + ∑ < , >) , (10) By the transformation formula (3) for connection matrices, the vanishing of a connection matrix is not an and = ∑ { ( ) + ∑ ( < , > invariant property. In fact, for an arbitrary connection, we + < , >) can always find a local frame field with respect to which ∑ ( < , > + ∑ < , > < , >)} . the connection matrix is zero at some point. This fact is useful in calculations involving connections. Hence − = ∑ { [ , ] + ∑ ( < Theorem 2. Suppose is a connection on a vector bundle [ , ], > + < , > − ∑ >)} = , and ∈ . Then there exists a local frame field in a [ , ] + ∑ , < , > (11) coordinate neighborhood of such that the corresponding connection matrix is zero at . That is,
Proof. Choose a coordinate neighborhood ( ; ) of ( , ) = − − [ , ] such that ( ) = 0, 1 ≤ ≤ . Suppose is a local This completes the proof of the theorem. □ frame field on with corresponding connection matrix = ( ), Theorem 4. The curvature matrix satisfies the Bianchi identity where = − . = Γ (7) Proof: Apply exterior differentiation [9] to both sides of = − = − + = −( + ) + ( + ) and the Γ are smooth functions on . Let = − = − Γ ( ). This completes the proof of the theorem. □ Remark 2. If a section of a vector bundle satisfies the Then = ( ) is the identity matrix at . Hence there condition = 0, then is called a parallel section. exists a neighborhood ⊂ of such that is non- III. Affine Connections degenerate in . Thus Definition 2. Let be a smooth n-dimensional manifold, = . (8) be the set of smooth functions and Γ( ) be the vector space is a local frame field on . Since of smooth vector fields. An affine connection on is a map (denoted by ) ( ) = − ( ), ∶ Γ( ) × Γ( ) → Γ( ) we can obtain from (3), ( , ) ↦ ( ) = ( . + . . )( ) such that = − ( ) + ( ) ( ) ( + ) = + = 0 ( ) = + Connections on Bundles 193
( ) ( ) = ( ) + Γ = + . (12) ( ) = ; ∀ ∈ and , ∈ Γ( ) under a local change of coordinates. Therefore the Γ are the IV. Affine Connection in Two Coordinates Charts coefficients of some connection and is torsion-free. Let ( , ) be a coordinate chart on a manifold , with Theorem 5. Suppose is a torsion-free affine connection on coordinates ( , , … , ). Then the vector fields and . Then for any point ∈ there exists a local coordinate can be expressed as system such that the corresponding connection coefficients Γ vanish at . = ( ) Proof. Suppose ( ; ) is a local coordinating system at with connection coefficients Γ . Let ( ) = 1 = + Γ ( )( − ( )) ( 2 For some smooth functions ( ) and ( ). In , − ( )) (13) are smooth vector fields. ∇ is again a smooth Then, = , = Γ ( ) (14) vector field. Thus Thus the matrix ( ) is non-degenerate near , and (13) = Γ provides for a change of local coordinates in a neighborhood For some smooth functions Γ ( ) . Here Γ ( ) is a of . From (12) we see that the connection coefficients Γ function. in the new coordinate system satisfy ⇒ = ∑ Γ ; where = , = ( ) Γ = 0 ; 1 ≤ , , ≤ and = This completes the proof of the theorem. □ Let us compute Theorem 6. Suppose is a torsion-free affine connection on = ∑ ∑ . Then we have the Bianchi identity:
= ∑ ( ∑ ) [By axiom ( )] , + , + , = 0 . = ∑ ∑ ( ) [By axiom ( )] Proof. From Theorem 4, we have ∑ ∑ = − , = ( ) [By axiom ( )] ∑ ∑ ( ) that is, = ( + ) [By axiom ( )] ∑ ∑ ( ) ∑ = ( + Γ ) The functions Γ ( ) are called coordinate symbols of the = Γ − Γ . affine connection . The vector field is often called covariant derivative of vector field along the vector Therefore field . , = Definition 3. If the torsion tensor of an affine connection − Γ − Γ = 0, is zero, then the connection is said to be torsion free. A torsion-free affine connection always exists. In fact, if where in the last equality we have used the torsion-free property of the connection. Hence the coefficients of a connection are Γ , then the set 1 ( , + , + , ) = 0 (15) Γ = Γ + Γ . 2 Now since the coefficients of (15) are skew-symmetric with respect to , , ℎ , we have Obviously, Γ is symmetric with respect to the lower indices and satisfies , + , + , = 0 This completes the proof of the theorem. □
194 Md. Showkat Ali et.al
V. Connection Compatible with Tensors (u1,u2 ,u3 ) (u2 ,u1,u3 ) (u2 ,u3 ,u1 )
Let M be a smooth manifold and be any tensor in M. (u ,u ,u ), Mostly this can be interested in the case when = g is a 3 2 1 semi-Riemannian metric tensor on M, i.e., is a non-- so that is anti-symmetric in the first and the third variables. degenerate [1] symmetric (2, 0)- tensor, or when is On the other hand symplectic form on M, i.e., is a non-degenerate closed 2-form [7], [9]. If is a connection in M, i.e., a (u1 ,u2 ,u3 ) (u3 ,u2 ,u1 ) (u2 ,u3 ,u1 ) connection on the tangent bundle TM, then we have (u ,u ,u ), naturally induced connections on all tensor bundles on M, 1 3 2 all of which is denoted by the same symbol . so that is symmetric in the second and the third variables. Definition 4. The torsion of is the anti-symmetric This concludes the proof. □ tensor Theorem 7. There exists at most one symmetric connection T (X ,Y ) X Y Y X [X ,Y ], which is compatible with a semi Riemannian metric. where [X ,Y ] denotes the Lie brackets of the vector fields Proof. Assume that g is a semi-Riemannian metric on M, and ~ X and Y; is called symmetric if T 0. let and are two symmetric connections such that ~ The connection is said to be compatible with is - - g g 0; for all p M , consider the map parallel, i.e., when 0. :T M T M T M p p p given by Establishing whether a given tensor admits compatible (X,Y, Z) g(t(X ,Y ), Z), connections is a local problem. Namely, one can use partition of unity to extend locally defined connections ~ where t is the difference . Since t is symmetric, then and observe that a convex combination of compatible connections is a compatible connection. In local is symmetric in the first two variables. On the other hand, is coordinates, finding a connection compatible with a given anti-symmetric in the last two variables tensor reduces to determining the existence of solutions ~ for a non homogeneous linear system for the Christoffel (X ,Y,Z) (X ,Z,Y) g( X Y,Z) ~ symbols of the connection. g( X Y,Z) g( X Z,Y) g( X Z,Y) It is well known that semi-Riemannian metric tensors ~ (X,Y,Z) (X ,Y,Z) 0. admit a unique compatible symmetric connection, called g g the Levi-Civita connection of the metric tensor, which can ~ be given explicitly in [4]. Uniqueness of the Levi-Civita By Lemma 1, 0, hence t =0, and thus . Hence completes the proof. □ connection can be obtained by a curious combinatorial argument, as follows. ~ …………….. Suppose that and are connections on M; their ~ 1. Arnold’s V.I., 1978. Mathematical Methods of Classical difference is a tensor, that is denoted by t Mechanics, Springer Verlag. ~ 2. Brickell, F. and Clark, R.S., 1970. Differential Manifolds: t(X,Y) X Y X Y, An Introduction,Van Nostrand Reinhold Company, London. where X and Y are smooth vector fields on M. If both 3. Carno, M.P. do, 1992, Riemannian geometry, Birkh auser, ~ Boston. and are symmetric connection, then t is symmetric 4. Chern, S.S. Chern, W.H., K.S., Law, Lectures on Differential ~ ~ Geometry. t(X,Y) t(Y, X) X Y X Y Y X Y X 5. Gupta, P.P. and Malik, G.S. 2000. Tensors and Differential [X,Y][Y, X ] 0. Geometry, Pragati Prakashan, Meerut, U.P., India. Lemma 1. Let U be a set and :U U U be a map 6. Islam, J. Chrish, 1989. Modern Differential Geometry for Physicists, World Scientific Publishing Co. Pte. Ltd. that is symmetric in its first two variables and anti-- 7. Kobayashi, S. and Nomizu, K., 1996. Foundations of Differential symmetric in its last two variables. Then is identically Geometry, Volume 1, John Wiley and Sons, Interscience, New zero. York. Proof. Let u , u , u U be fixed. We have 8. Novikov, S.P. and Fomenko. A.T., Basic Elements of Differential 1 2 3 Geometry and Topology. 9. Struik, D.J., 1950. Lectures on Classical Differential Geometry, Addition-Weslely Publishing Co., Inc.