It seems that there is no digital copy of Leon Karp's Ph.D. thesis

L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.

on internet and his paper excerpted from his thesis is very brief and without any detailed proof. (I wonder that peers read the thesis or they trust to the advisory committee).

Karp, Leon, Parallel vector fields and the topology of manifolds, Bull. Am. Math. Soc. 83, 1051-1053 (1977). ZBL0376.53024.

There he generalized a theorem of S. S. Chern and proved that

Theorem. If $M^n$ admits a vector field that is parallel with respect to some Riemannian metric then the Betti numbers of $M$ satisfy $b_{k+1}\geq b_k-b_{k-1}$ and $b_1\geq 1$.

There is also no review on zbMATH. I want to know about its sketched proof; and similar results on bivectors if exist any.