By Leonard Lovering Barrett
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Extra info for An introduction to tensor analysis
6. One just has to require the total space to be a complex manifold and the ﬁber bundle charts to be holomorphic. In particular, one has the subclass of holomorphic vector bundles among complex vector bundles. For principal bundles with structure group a complex Lie group, there is the subclass of holomorphic principal bundles. Given any holomorphic ﬁber bundle, there is a natural notion of holomorphicity for sections via holomorphicity in some (or equivalently any) ﬁber bundle chart. 8, and so on.
Choosing a local smooth section σ of P, we get s(x) = σ(x), f (σ(x)) , which immediately implies smoothness of s. Conversely, any element in the ﬁber over p(u) may be uniquely written in the form u, y , so given s, the equation s(p(u)) = u, f (u) can be used to deﬁne f . Smoothness of f follows easily by writing this in terms of a local smooth section of P, while equivariancy is an immediate consequence of u·h, f (u·h) = u, f (u) . In the case of associated vector bundles, we can generalize this result to a description of diﬀerential forms with values in an associated bundle.
The Lie bracket of two elements of h again lies in h. e. [X, Y ] ∈ h for any X ∈ h and Y ∈ g. Given a Lie group G with Lie algebra g, there may exist Lie subalgebras h ⊂ g for which there is no Lie subgroup H ⊂ G having h as the tangent space at the identity. This can be seen from the case of the real torus T2 = S 1 × S 1 . The Lie algebra of this is R2 with the trivial Lie bracket, so a subalgebra is just a linear subspace. Now if one takes a line of irrational slope, then it is easy to see that any subgroup H ⊂ G, which contains a small submanifold around the unit element that is tangent to the given line, must be dense in G, so it cannot be a Lie subgroup.