# Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)

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.

• Have you tried to email Karp himself? – Deane Yang Apr 17 at 16:30
• No, he is active still? – C.F.G Apr 17 at 16:33
• I suggest emailing him to see if he's willing to help you with this. These days, if you can't find something like this elsewhere, contacting the author is relatively easy to do. – Deane Yang Apr 17 at 16:40
• There's no guarantee of a response, but it doesn't hurt to try. – Deane Yang Apr 17 at 16:53
• The standard source for all North American PhD dissertations is ProQuest (previously known as University Microfilms). Indeed, a single search immediately revealed that they do have a copy of the dissertation under discussion, which I was able to download and read. I posted a link for your convenience. – Dmitri Pavlov Apr 17 at 20:11

The full text of Karp's thesis (a scanned PDF file) is available here:

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

• Thanks. Did you download the 51 pages version or just previewed 12 pages? ProQuest says it cost \$41!! – C.F.G Apr 17 at 21:06
• @C.F.G: You need a subscription to download the full text. And if your university does not have one, an interlibrary loan service will get it for you. But I also posted the full text here: dmitripavlov.org/scans/karp.pdf – Dmitri Pavlov Apr 17 at 21:42