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Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in $\mathbb{R}^n$. Given a manifold $M$ of dimension $m$, are there any methods to study whether $M$ can be embedded into $G_k(\mathbb{R}^n)$ and $AG_k(\mathbb{R}^n)$?

When studying the embedding/immersion problem of manifolds into Euclidean spaces, I learned that there are some obstructions by Stiefel-Whitney class from lecture notes. How about embedding problems into Grassmannians?

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    $\begingroup$ Stiefel whitney classes give obstructions for embeddings $f:M\rightarrow N$ for arbitrary manifolds as we have an exact sequence of vector bundles $0\rightarrow TM\rightarrow f^*TN \rightarrow NN\rightarrow 0$. But the problem is now more delicate because $f^*w(TN)$ is not known (even if the SW classes of TN are given). $\endgroup$
    – Thomas Rot
    Oct 21, 2015 at 11:46
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    $\begingroup$ I have not done any calculations for the Grassmannians but one should be able to glance information for this sequence. $\endgroup$
    – Thomas Rot
    Oct 21, 2015 at 11:54
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    $\begingroup$ I suppose you should pose a more specific question. In this generality, you can use the Whitney embedding theorem to embed a compact $m$-manifold into the Grassmannian whenever $2m\leq k(n-k)$. Probably you do not want this embedding, but embeddings whose homotopy class in $[M,G_k(\mathbb{R}^n)]$ satisfies some conditions, or embeddings with significantly sharper bounds... There are things like h-principles for immersions or Weiss' embedding calculus, but without further information it is difficult to know if these things help. $\endgroup$ Oct 21, 2015 at 12:34
  • $\begingroup$ @ThomasRot Thanks, Thomas! In your comment, since $N$ is not assumed to be embedded in an ambient Riemannian manifold, how to define $NN$? Does it mean the normal bundle of $N$? Where can I find the exact sequence? $\endgroup$
    – Shi Q.
    Oct 21, 2015 at 13:02
  • $\begingroup$ Yes it means the normal bundle. If you want you can take a Riemannian metric on $TN$ and take the complement and pull it back to $M$. $\endgroup$
    – Thomas Rot
    Oct 21, 2015 at 13:08

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