An Introductory Course on Differentiable Manifolds by Siavash Shahshahani

Author:Siavash Shahshahani [Siavash Shahshahani]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2017-12-08T16:00:00+00:00

From the linearity of β in the first component, we infer that in β♭(u)∈F*, and from the linearity in the second component, that β♭ is linear, β is called non-degenerate in case β♭ is an isomorphism. This is equivalent to β♭(u) not being identically zero, unless u=0. Inner products are examples of non-degenerate bilinear maps. It is useful to have an explicit expression for β♭ in terms of β. Suppose B=(e1, . . . , ek) is a basis for F, and B*=(e1, . . . , ek) is the dual basis for F*. We know from Chapter 1 that β can be written as . Since β♭(ej)∊F*, we may write . Therefore, Cij = (β♭(ej))(ei) = β(ei, ej) = βij. It follows that the matrix of β♭:F→F* with respect to the bases B and B* for F and F*, respectively, is B=[βij]. In the non-degenerate case, the inverse of β♭ is denoted by β♯. Therefore, the matrix of β♯, relative to the bases B* and B, is given by [βij], where [βij]=B-1. We summarize these results in the following lemma for future reference.

19. Lemma Let F be a finite-dimensional vector space ove with ordered basis B=(e1, . . . , ek), and let B*=(e1, . . . , ek) be the dual basis for F*. If is a covariant 2-tensor on F, then the matrix of ß♭ relative to B and B* is given by B=[βij]. If further, β is non-degenerate, then the matrix of β♯ relative to B* and B is given by the inverse B-1=[βij].

Now consider a Cr vector bundle (E, π, M) of fiber type F, and suppose that instead of a single bilinear map, we have a Cr cross-section of E⊕E. For each x∈M, we obtain a linear map β(x)♭:Ex→. Thus a map β♭:E→E* is obtained that sends the fiber Ex to the fiber , for each x. The following commuative diagram summarizes this:

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