I am a second year math major at UChicago, and I plan to write some blog posts here on complex geometry as a way of helping myself learn the material. I might also write some posts on other stuff.
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First let us start with the case of a smooth manifold of real dimension . Let us look at a tangent vector of at the point . It means the following. We take a curve with . In local coordinates is given by . The tangent vector is given by a column vector whose components are . If we use another local coordinate system , then the tangent vector is given by a column vector with components . By the chain rule the two column vectors and are related by . Formally the transformation rule can be described by . Another way of looking at the formal expression is that one knows a tangent vector if and only if one knows how to take the partial derivative of any function in the direction of the tangent vector, because the components of the tangent vector are precisely the partial derivatives of the coordinate functions along the tangent vector. The formal expression is simply the partial differential operator in the direction of the tangent operator.
The set of tangent vectors form a bundle over in the following sense. Suppose is covered by coordinate charts with coordinate . Through the coordinate chart we identify all tangent vectors at points of by column vectors . So the totality of all tangent vectors at points of is given by . So the totality of all tangent vectors at points of is obtained by taking the disjoint union of for all and then identify with by , where is the th component of . So we have a bundle with transition functions . This is the tangent bundle and we denote it by .
Suppose now we have a tangent vector field on some open subset of . That is, we have a section of the tangent bundle over . We would like to be able to differentiate the tangent vector field along some given direction and get a tangent vector field. This is a question of how to differentiate the section of a vector bundle and get a section as a result. We would like to tackle this question in the general setting. Suppose we have a vector bundle of rank with transition functions with respect to a covering . A section over an open subset is given by so that is an -tuple of functions over and . Here no summation is used and are column vectors and is an matrix. For our differentiation we do not specify the direction and consider the total derivative. From we have . If the term is not present, we get a section of from after we specify a direction. However, in general we have the term and it is not possible to use for differentiation and get a section. We have to use other ways to differentiate sections of a bundle.
We want the procedure of differentiation to obey the Leibniz formula for differentiating products. So to define differentiation it suffices to define differentiation for a local basis of . We denote the (total) differential operator by . Then is a -valued 1-form and we can express it in terms of our local basis and get , where is a 1-form. To be able to differentiate sections is equivalent to having an matrix valued 1-form . This matrix valued 1-form depends on the choice of . Suppose we have another local basis which is related to by . Then
In matrix notations we have , where are column vectors with components and and . Thus and . So to be able to differentiate sections is equivalent to having an matrix valued 1-form for any local basis so that the transformation rule is satisfied. In that case we say that we have a connection.
When we have a connection for , we have an induced connection on the dual bundle of . It is given as follows. Suppose is a local section of and is a local section of . Then is defined so that is satisfied, where means the evaluation of at . If is the dual basis of , then the connection is simply equal to .
Suppose we have an inner product along each fiber of . We say that a connection is compatible with the metric if for any section over a curve in with along the length is constant along . This condition is equivalent to . Clearly this equation implies that if along then is constant along . Conversely, for a point and a tangent vector at we can find a curve passing through with tangent vector at . We choose a frame field along so that the frame field is orthonormal at and each has zero covariant derivative along . Then the frame field is orthonormal at each point of . Write and . Then evaluated at ,
By applying to belonging to a local orthonormal basis, we conclude that compatibility with the metric is equivalent to the connection for an orthonormal basis is skew-symmetric.
Now we come back to the tangent bundle of . An inner product along the fibers of is simply a Riemannian metric. We introduce a connection so that we can differentiate sections of and get sections again. having a connection for is equivalent to having a connection for the dual bundle of . A section of is simply a 1-form. For a 1-form we have always the concept for exterior differentiation. We do not need any connection for exterior differentiation. The result of an exterior differentiation of a 1-form is a 2-form . It is different from coming from a connection, because is a section of whereas is a section of . We would like to consider connections that relate to in a natural way. A connection is said to be torsion-free if the skew-symmetrization of is . Suppose is a local frame for and is its dual frame. Let with be the matrix valued 1-form of the connection. Then from definition of the torsion-free condition we see that the connection is torsion-free if and only if .
Fundamental Theorem of Riemannian Geometry. There exists a unique torsion-free connection that is compatible with a Riemannian metric.
Proof. Let the Riemannian metric be given by . We use as local basis for and as local basis for . Write the connection for as . The coefficients are known as the Christoffel symbols. So . The torsion-free condition is simply the symmetry of in and . Compatibility with the Riemannian metric means
To simplify notations we let and let . The theorem is reduced to proving the existence and uniqueness of symmetric in and which satisfy . This is a linear algebra problem. We have unknowns and as many equations. It is in general difficult to handle such a large system of linear equations. Fortunately this large system can be decoupled into smaller sets of three equations in three unknowns. For a fixed triple , since is symmetric in and , by permuting we have only three unknowns , , and . Such permutations would generate from three equations. We can now solve uniquely our three unknowns from these three equations. The three equations are
The usual way is to add up the three equations so that we know the sum of the three unknowns and then subtract from it the sum of two unknowns obtained from any of the three equations. This is equivalent to subtracting one equation from the sum of the other two. So we subtract the third equation from the sum of the first two equations and we get
The unique connection is now given by
This unique connection is called the Levi-Civita connection or the Riemannian connection.
We would like to introduce the invariant formulation for the torsion-free condition. In the literature the standard notation for the covariant differential operator is instead of . We will interchangeably use both and to denote the covariant differential operator. The torsion-free condition given by the symmetry of the Christoffel symbols in the two covariant indices when expressed in therms of local coordinates. The Christoffel symbols are given by and . In other words, . The torision-free condition is that is symmetric in and , i.e. is symmetric in and when and equal and . For general vector fields and we do not have from the torision-free condition, because for smooth functions and ,
and we cannot expect to get the general case by multiplying the special case , by smooth functions and then summing up. The trouble is the discrepancy terms . The Lie bracket yields the same discrepancy terms when and are respectively multiplied by smooth functions and , as follows.
So we conclude that the torsion-free condition is equivalent to the vanishing of for any pair of tangent vectors and . Moreover, we have seen that
When the torsion-free condition is not satisfied, we define . Then and is a tensor and is called the torsion tensor. In local coordinates write with .
We give here a geometric interpretation of torsion. Take a point of and two tangent vectors and at . We take a curve going through whose tangent at is . We transport for a distance along and get a tangent vector of at . Take a curve through whose tangent at is . Let be the point . Now reverse the roles of and . We take a curve going through whose tangent at is . We transport for a distance along and get a tangent vector of at . Take a curve through whose tangent at is . Let be the point . The limit of the vector (in any coordinate system) as and approach zero defines a tangent vector of which is independent of the choice of the curves by order considerations. We claim that this tangent vector is . Let us now verify. We choose a local coordinate system at so that corresponds to the origin. Let and be respectively the components of and in terms of this local coordinate system. We choose as our curve the curve defined by the equations . The equation for parallel transport of along is . Hence . Here higher orders are ignored in the computation. We can take as the curve defined by the equations . Hence . Likewise we get the expression for by changing the roles of , and , . Thus .