Thread: Lebesgue measure of a manifold embedded in the euclidean space

Let M be a m-dimensional manifold embedded in a n-dimensional euclidean space. If m is strictly less than n, is its Lebesgue measure zero?

Can anyone give me a reference which can prove/disprove this?

Thanks

Originally Posted by uasac

Let M be a m-dimensional manifold embedded in a n-dimensional euclidean space. If m is strictly less than n, is its Lebesgue measure zero?

Can anyone give me a reference which can prove/disprove this?

Thanks

Here's how I would think about doing it. Since can be embedded into and subspaces of Lindelof spaces are Lindelof we know that is Lindelof and so we can find a countable atlas for . Since the countable union of null sets is null it suffices to show that each has measure zero. To do this consider that by considering the map (where is the embedding ) you get a smooth map from into and since we know (by lemma 10.3 in Lee) the image of has measure zero. And, since measure zero sets are preserved under diffeomorphisms we can pull back along to get that has measure zero.

I'm just starting to learn differential geometry, but your arguments seem sound to me. Thank you.