Nonexistence of Almost Complex Structures on Grassmann Manifolds

Abstract

Abstract.In this paper we prove that, for 3 < k < n -3 , none of the oriented Grassmann manifolds, Gn k-except for G6 3 , and a few as yet undecided cases-admits a weakly almost complex structure. The result for k = 1,2, n -1, «-2 are well known and classical. The proofs make use of basic concepts in iT-theory, the property that Gn k is (n -fc)-universal, known facts about K(RP ), and characteristic classes.

Nonexistence of Weakly Almost Complex Structures on Grassmannians

“…Our results are sharper than that in [6]. It is well known that the tangent bundle TGk(Rn) (TGk(Rn)) of Gk(Rn)) has the following description (see [4]): …”

“…The statement that G2(R 4) and G3(R 6) are weakly almost complex was obtained in [6]. The statement that G2(R 4) and G3(R 6) are weakly almost complex was obtained in [6].…”

Almost complex structures on quaternion-Kähler manifolds and inner symmetric spaces

“…Note however that this was already known for HP n , as mentioned above, and also for most real oriented Grassmannians Gr 4 (R n+4 ). Indeed, the nonexistence of (weakly) complex structures on a large class of real Grassmannians, including all Gr 4 (R n+4 ) except for Gr 4 (R 8 ) and Gr 4 (R 10 ), was shown in [24] by P. Sankaran and in [27] by Z.-Z. Tang.…”

“…The non-existence of weakly complex structure for all oriented real Grassmannians, except for Gr 4 (R 8 ), Gr 6 (R 12 ) and Gr 4 (R 10 ), in particular for all oriented real Grassmannians of dimensions 4n+2, was established, by different methods, by P. Sankaran in [24] and by Z.-Z. Tang in [27].…”

“…In the above list, these are: (i) Hermitian symmetric spaces of odd complex dimension; (ii) oriented real Grassmannians Gr 2p (R 2p+q ), whith p and q both odd; (iii) the exceptional symmetric space E 7 SU(8)/Z 2 (which is of dimension 70). The non-existence of weakly complex structure for all oriented real Grassmannians, except for Gr 4 (R 8 ), Gr 6 (R 12 ) and Gr 4 (R 10 ), in particular for all oriented real Grassmannians of dimensions 4n+2, was established, by different methods, by P. Sankaran in [24] and by Z.-Z. Tang in [27].…”