Geodesic
The incidence submanifold of $RP^n\times G_1(RP^n)$ for $n$ odd is nonorientable
Mathematica slovaca, Tome 26 (1976) no. 2, pp. 101-102
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Classification : 57M35, 57N65 
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     author = {Ru\v{z}ick\'y, Eugen},
     title = {The incidence submanifold of $RP^n\times G_1(RP^n)$ for $n$ odd is nonorientable},
     journal = {Mathematica slovaca},
     pages = {101--102},
     year = {1976},
     volume = {26},
     number = {2},
     mrnumber = {0431238},
     zbl = {0319.55012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MASLO_1976_26_2_a3/}
}
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Ružický, Eugen. The incidence submanifold of $RP^n\times G_1(RP^n)$ for $n$ odd is nonorientable. Mathematica slovaca, Tome 26 (1976) no. 2, pp. 101-102. http://geodesic.mathdoc.fr/item/MASLO_1976_26_2_a3/

[1] DOLD A.: Lectures on Algebraic Topology. Springer-Verlag 1972. | MR | Zbl