Geodesic
Pontryagin manifolds
Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 441-447

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The Pontryagin manifold $P_{n,k}$ is the set of $(k+1)$-frames in $\mathbf R^n$ such that the dimension of the linear span of the vectors in the frame is no less than $k$. In the theory of Pontryagin classes these manifolds play a role analogous to that of the Stiefel manifolds in the theory of Stiefel–Whitney classes. The present paper examines the homotopy type of these manifolds. The results are then applied to study the connection between immersions and $k$-immersions. Bibliography: 8 titles. 
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     author = {N. V. Ivanov},
     title = {Pontryagin manifolds},
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N. V. Ivanov. Pontryagin manifolds. Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 441-447. http://geodesic.mathdoc.fr/item/SM_1980_36_3_a9/