Any set of vectors in some Euclidean space such that any pair of them lie at an obtuse angle are either linearly independent or have a linear combination with non-negative coefficients equal to zero.
Let and be complete flags over an -dimensional vector space . Show that there is a set of lines compatible with the two flags, in the sense that there exists a permutation such that and .
Let be a Lie algebra and its Killing form. Is the kernel of always equal to the radical of ?
Let be a connected Lie subgroup of . Then commutes with all of iff it commutes with all of .
Is there a compact 3-manifold having as boundary?
A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.
Prove the following two statements and conclude that the fundamental group of a (connected) topological group is abelian:
1) A discrete normal subgroup of a connected topological group is central.
2) If is the universal covering, the total space admits a topological group structure such that is a group morphism. The kernel of is then isomorphic to the fundamental group of .