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**](https://doi.org/10.1090/S0002-9904-1977-14379-6), Bull. Am. Math. Soc. 83, 1051-1053 (1977). [ZBL0376.53024](https://zbmath.org/?q=an:0376.53024), [MR451257](https://mathscinet.ams.org/mathscinet-getitem?mr=451257).


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](https://www.zbmath.org/). I want to know about its sketched proof; and similar results on bivectors if exist any.



Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
The full text of Karp's thesis (a scanned PDF file) is available here:


<https://search.proquest.com/pqdtglobal/docview/302809402/>


