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In Fibre Bundles (3rd Ed.) by Husemoller, the canonical $k$-dimensional bundle $\gamma^n_k$ on $\mathrm{Gr}_k(\mathbb{R}^n)$ is introduced as the subbundle of the product bundle, $\mathrm{Gr}_k(\mathbb{R}^n) \times \mathbb R^n \overset{p}{\to} \mathrm{Gr}_k(\mathbb{R}^n)$ defined by the fact that the total space of $\gamma^n_k$ is all pairs $(V,x) \in \mathrm{Gr}_k(\mathbb{R}^n) \times \mathbb R^n$ such that $x \in V$.

I am having trouble connecting this with the definition of the canonical bundle as the exterior power of a line bundle, and my question is essentially how are both of these related in this case?

Furthermore, is it possible to formulate the general definition of the canonical bundle using Husemoller's explanation, i.e. as a subbundle of a product bundle, rather than exterior powers?

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    Does Husemoller really use the term 'canonical bundle'? That bundle is usually called the [*tautological* bundle](https://en.wikipedia.org/wiki/Tautological_bundle). – 2017-01-24
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    @MichaelAlbanese Yes, "the canonical $k$-dimensional vector bundle" are his exact words. – 2017-01-24
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    That's uncommon. The two notions are unrelated, and that's probably why the name 'canonical bundle' for the bundle over a Grassmannian didn't catch on. – 2017-01-24
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    @MichaelAlbanese Damn, I hope there isn't any other misused naming/notation in the book. – 2017-01-24
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    @MichaelAlbanese Ok, I have a question for you: Do you consider wikapedia to be the standard of correct mathematical terminology ? – 2017-01-24
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    @ReneSchipperus: Not necessarily. – 2017-01-24

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I don't know Husemoller's book, but that usage of "canonical" is nowadays more commonly called "tautological"; that is, the tautological bundle of $k$-planes over the Grassmannian has fibre over $V$ equal to $V$ itself.

The top exterior product of the cotangent bundle is related only by name.

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    Whoops...Michael beat me to it. Wikifying.... – 2017-01-24
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    In future, don't worry about it. I am sure you know what I know and a lot more, so I won't think you're copying. – 2017-01-24
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    @MichaelAlbanese: To the contrary, there's surely a lot of mathematics you know that I don't. :) If my answer had contained more information than your comment I might not have wikified, but under the circumstances it seemed best here. – 2017-01-24