We now look at the case of holomorphic vector bundles over a complex manifold and discuss the concept of second fundamental form. Now, is a complex manifold of complex dimension , and is a holomorphic vector bundle of rank over . Assume that has a Hermitian metric. Let be a holomorphic vector subbundle of of rank .
We can choose a local unitary basis of such that belongs to . Let be the connection of induced by the complex metric connection of . In other words,
for . Another way of describing the connection is that the -covariant derivative of a local section of of is obtained by taking its -covariant derivative and then projecting onto by the orthogonal projection. This connection agrees with the complex metric connecion of with respect to the metric induced from that of . The reason is as follows. Firstly, it is easy to see that the -covariant derivative of a local holomorphic section of along any direction is zero from the above description of the -covaraint differentiation. Secondly, if is a section of above a local curve of with zero -covaraint derivative along the curve, then its -covariant derivative is a linear combination of and must be perpendicular to . As a consequence,
must vanish along the curve, and the length of is constant. Let be the orthogonal complement of in . We give the complex structure of the quotient bundle . The difference of the -covariant derivative of and the -covariant derivative of of is a 1-form with values in . This -valued -form is called the second fundamental form of in , and we denote it by . Let
be the representation of in terms of some local holomoprhic atlas . Then
is a -valued -form. Thus, the second fundamental form must be a -valued -form, and we can write
for a section of .
Let us now consider the case of the quotient bundle. Take a local holomorphic basis of . We use the same notation to denote local sections of orthogonal to . The holomorphicity of basis means that for some sections of , the sections are holomorphic sections of . We take also a local holomorphic basis of . We have
When we write
in terms of the local holomorphic basis of , we see that
Hence, is an -valued -form. Since from the definition of one has
it follows that is a -valued -form. So we can define a connection on by
We claim that this connection is the complex metric connection of . It is a complex connection, because we have observed earlier that
and the -form has values in . It is also a metric connection, because
for sections of . The operator is a -valued -form given by
We call the second fundamental form of the quotient bundle of .
Another more invariant way of representing the second fundamental form of the quotient bundle of is the following. We have
The local basis of the orthogonal complement of in simply describes the monomorphism from to which lifts to the orthogonal complement of in . let us call this monomorphism . Then is simply . The entity is a priori only a -valued -form which is exact. Our previous discussion shows that it is actually a -valued -form. However, as a -valued -form, in general it is only closed and is not exact. Suppose we have another Hermitian metric for . Then we would have a different monomorphism from to and a different second fundamental form . The difference of and is a homomorphism from to , because and are different liftings of to . So the difference of the two second fundamental forms of equals , which is a exact -valued -form.
We want to relate to the second fundamental form of the subbundle of . For sections and of and , respectively, we have
Hence, is simply the negative of the adjoint of with respect to the Hermitian metrics of and . So
We now compute the curvatures of , , and . We choose a local orthonormal frame so that belongs to . Write
Since the connection is compatible with the metric by differentiating
we conclude that
The second fundamental form of is given by
for . From
(no summation over ) for any vector of type , and equality holds if and only if
In invariant formulation, we have
for any in and any vector of of type .
Let us now look at the case of the quotient bundle. We can identify the quotient bundle as the subbundle of which is the orthogonal complement of in . We needed the complex structure of only to define the connection of as a complex metric connection. Once we get the connection of , we can ignore the complex structure of . The calculation of the curvature tensor depends only on the connection. So when it comes to comparing the curvature tensors, the computation of the quotient bundle case is the same as the subbundle case. There is however one difference. The second fundamental form of the subbundle is an endomoprhism valued -form. So when it comes to evaluating the exterior product of the second fundamental form and its complex conjugate transpose at for some -vector , there is a sign difference between the quotient bundle case and the subbundle case. So we have
for any in the orthogonal complement of in and for any vector of of type , where is the image of in .