Suppose is a complex manifold of complex dimension . We can regard as a real manifold of real dimension . For local holomoprhic coordinates , we write . Then and . The complexification of splits into a direct sum , with spanned by and spanned by . We have a natural map from to , which is the composite of the inclusion map and the projection map onto the first summand. This natural map is an isomorphism, and its inverse is the map from to . Through the -isomorphism , the operator of multiplication given by in corresponds to some operator in . From and , it follows that the operator is given by and . This means that if is made into a -vector space by defining multiplication by as , then is a -isomorphism between the -vector spaces and . We identify and through this isomorphism. Note that
and by analogous direct derivation or from the fact that is real, one has . This simply means that and are respectively the eigenspaces of for the eigenvalues and .
Since with the operator becomes a -vector space, when we have a metric on to measure the length of the vectors on , we can ask whether this length function comes from a Hermitian inner product. One necessary condition is that the length of a vector should be invariant under . It turns out that this condition is also sufficient. So we are going to use this condition as definition of a Hermitian metric and then verify that it agrees with the usual notion of a Hermitian metric. We say that a Riemannian metric on is Hermitian if for any , in . Now we are going to reconcile this definition with the usual definition of a Hermitian metric respresented by a Hermitian matrix. A Riemannian metric is a real-valued -billinear function on . We can extend it by -bilinearity to a -bilinear function on . Then the Riemannian metric is broken into four parts , , , . We investigate the condition for each part. For the part , since is the same as multiplication by , we have , and by -bilinearity, we have . From an analogous argument or from the fact that is real, we have .
Since is symmetric, we have . Moreover, since is real, we must have , i.e. . Hence , and is a Hermitian matrix. So the condition simply means that , is Hermitian, and is the transpose of .
The correspondence between and transports the metric of to a metric on , and we want to determine this metric on . For this metric on , the inner product of and is
So the Riemannian metric on correpsonds to the metric on . If we only look at the length of a vector, the metric on is simply the Hermitian metric .Let us now explain the difference between and . A metric is used to measure the length of a vector. In that sense, and agree. The difference comes in when we want to express the square of the length of the vector as its inner product with itself. In the case of an -vector space, the inner product should be an -bilinear functional. In the case of a -vector space, the inner product should be a functional which is -linear in the first argument and -conjugate linear in the second argument. When the lengths of the vectors are determined, in either case there is only one way of writing out such an inner product, and it is done by polarization. When we regard has an -vector space, we have the inner product for the metric, whereas when we regard as a -vector space, we have the Hermitian inner product for the metric.
If one wants to keep working only with real vectors all the time, one can verify directly in the following way that the condition implies that the length function defined by comes from a Hermitian inner product on when is made into a -vector space by the operator . The map is a -isomorphism between and . The inverse of this map is because . Hence we expect the Hermitian inner product to be when is extended by -bilinearity. In other words, the Hermitian inner product is
which is the same as . This procedure is the same as getting the Hermitian inner product from the square of the length by polarization. We would like to note that is the real part of the Hermitian inner product, and is its imaginary part. In terms of abstract algebra, the above argument can be formulated in the following way. An -bilinear inner product on a -vector space is the real part of a Hermitian inner product if and only if it is invariant under the operation of multiplication by .
Let us now consider the Levi-Civita connection expressed in terms of the complex coordinate indices , . We let denote both and . We have
Suppose . Then must be of type to make a nonzero contribution. So
These Christoffel symbols are the connection of , which is induced by the Levi-Civita connection of . In general, this connection of does not define a connection of , because may not be zero. So when we differentiate a section of in the direction, we may get a section of which is not entirely in . We get a connection of if and only if vanishes.
We want to get a connection of from through the -isomorphism . In the correspondence between and , when we take a section of , we should consider in , which must be of the form . Then in is equal to . So when we take , we should be looking at
Here , go through the range . Thus in , we have
So the connection is complex if and only if The induced connection of is automatially compatible with the metric because the Levi-Civita connection of is compatible with the metric and the length functions of the metrics of and correspond through . Moreover, when the connection is complex, the connection is given by . The formula for the Christoffel symbol of the Levi-Civita connection gives
We note that this connection is the same as the complex metric connection for the holomorphic vector bundle with the Hermitian metric (or ). The Hermitian metric is called Kähler if the Levi-Civita connection is complex. An equivalent condition is that . This is equivalent to the condition that the -form is closed. The -form is equal to
because is and . The function is skew-symmetric in and because . So defines a -form on . Consider the Hermitian form . Twice its imaginary part is , which is . Thus the -form is the imaginary part of , and the Riemannian metric is twice its real part. We have also seen this before. The -form is called the fundamental form of the Hermitian metric. So a Hermitian metric is Kähler if and only if its fundamental form is closed.
We now look at another characterization of the Kähler condition for a Hermitian metric. This characterization is that for any point of , there exists a local holomorphic coordinate system centered at so that the first derivative of vanishes at . Clearly if such a coordinate system exists, then the fundamental form of the Hermitian metric is zero. Conversely suppose the metric is Kähler. Without loss of generality, we can assume that equals the Kronecker delta at the point . We seek a new coordinate system which is related to by , with symmetric in and . Let . Since is symmetric in and , it follows that
and at equals
Thus we should choose . This can be done with symmetric in and if and only if is symmetric in and .
We would like to remark why in the real case one can always find local coordinates so that the first derivative of the coefficients of the metric tensor vanishes at a point. In the real case, we want to vanish at . In other words, with symmetric in and . Recall the equations which were used to solve for the Christoffel symbols in terms of the Riemannian metric. All we have to do is set . This solution is the same as getting a coordinate system by integrating out along geodesics emanating from .
Another characterization of the Kähler condition for a Hermitian metric is that all the Christoffel symbols for the Levi-Civita connection vanish except the types and its complex conjugate . We have seen that always vanishes. We have the vanishing of if and only if the Hermitian metric is Kähler. We get our conclusion because is symmetric in and , and it is real in the sense that . Geometrically, this characterization of a Kähler metric says that a Hermitian metric is Kähler if and only if the covariant derivative of a vector field of type is still of type . In other words, types are preserved under parallel transport. This is equivalent to our earlier observation that vanishes if and only if the connection of induced from the Levi-Civita connection of defines a connection of through the inclusion map .