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I tried to find some paper published in Notices of the American Mathematical Society 1975
By E. Calabi, On manifolds with non-negative Ricci curvature II
Notices of the American Mathematical Society 22 1975 A205

But when I searched at mathscinet and click Calabi, there is no such a paper listed under his name.
Also AMS's website does not have any issues from before 1995. So is it a book or something?

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The earliest Notices entry I was able to get from MathSciNet was 1983. I'm not quite sure why. –  Thierry Zell Nov 12 '12 at 22:42
Long ago, the Notices of the AMS published essentially just meeting announcements and abstracts of talks. The Calabi citation seems to be well within this "long ago", so it is probably just an announcement of results, not a real paper. –  Andreas Blass Nov 12 '12 at 22:47
@Andreas Blass, I see, thank you. That's probably the reason why no actual pages was referred in the paper. –  J. GE Nov 12 '12 at 22:51
As Andreas points out, the Notices has evolved from being just a formal record of meetings (with abstracts), elections, etc. Among the abstracts published were quite a few that didn't correspond to talks actually given at meetings and that may or may not show up in published work later on. At that time the Bulletin published more detailed multi-page research announcements, but these too didn't always have follow-up. –  Jim Humphreys Nov 13 '12 at 0:46
Yes, that "A" part of the number means it is the listing of an abstract. –  Gerald Edgar Nov 13 '12 at 1:14

1 Answer 1

up vote 19 down vote accepted

As Andreas Blass and others surmised, this is indeed an abstract. It was for a 10-minute talk at the Annual Meeting in January, 1975. Here is the entire "paper":

On Manifolds with nonnegative Ricci curvature II

Let $M$ be an $n$-dimensional Riemannian manifold with nonnegative Ricci curvature. Then the exponential mapping $\exp_p$ for any $p\in M$ , restricted to the domain bounded by the cut locus, is everywhere volume decreasing.From this fact one deduces the following THEOREM. Let $M$ be a Riemannian, $n$-dimensional, complete manifold with nonnegative Ricci curvature. Then, if $r$ denotes the injectivity radius and $D$ the geodesic diameter of $M$ , the volume $V$ of $M$ satisfies $V \ge c_nr^{n-1}D$, where $c_n$ is a positive constant depending on $n$ . In particular, if $M$ is not compact (i. e. if $D=\infty$), the volume of $M$ , under the same assumptions, is infinite. (Received November 6, 1974.)

A couple of notes. I reproduced the capitalization and (non)hyphenation of the title as it appeared in the Notices. I also tried to preserve some oddities in punctuation in the text, but otherwise "TeX-ified" it; the original is literally typed, with a handwritten $\in$ symbol.

Added 11/13/12: Out of idle curiosity, I went back to the library today, to see if Calabi ever gave a talk titled "On Manifolds with nonnegative Ricci curvature I." If he did, it wasn't at an AMS meeting (or else I didn't dig back far enough).

In the process, however, I noticed that, beginning in October, 1972, the Notices ran a "Queries" column, inviting "questions from members regarding mathematical matters such as details of, or references to, vaguely remembered theorems, sources of exposition of folk theorems, or the state of current knowledge concerning published conjectures" -- i.e., a sort of snail-mail version of MathOverflow. Here's the inaugural query (the answer to which arrived in a then-speedy three months):

1 . R.P. Boas (2440 Simpson Street, Evanston, Illinois 60201). Given a finite collection of vectors, of total length 1, in a plane, we can always arrange them in a polygon, starting from 0, that at some stage gets at least $1/\pi$ away from 0. Mitrinovic [Analytic inequalities, 1970, pp. 331-332] cites Bourbaki [1955], but the theorem was known at least in the early 1940's, when I remember seeing a paper on it; can anybody supply the reference?

My own favorite is from the next issue:

4 . Cleve B. Moler (Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87106). Can somebody recommend a good source where I can learn about the connection of mathematics and various biological processes such as photosynthesis?

Cleve Moler is perhaps best known as the inventor of MATLAB.

In conclusion, the answer to the OP's question, "is the Notices available before 1995?" the answer seems to be yes, but only at libraries that hold onto old journals. I wonder if the AMS could be persuaded to make the early volumes of the Notices available through JSTOR.

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This result was proved almost at the same time by Yau, his proof appeared here (Theorem 7). –  YangMills Nov 13 '12 at 4:02
@YangMills, I knew Yau's proof and was trying to find what was the "other method" that Calabi had in his mind. But thanks anyway. –  J. GE Nov 13 '12 at 9:29
@Barry: My personal copies have long sincd been recycled, but intrepid people can actually find these back issues locally. Not in the UMass library (or nearby small college libraries), but under a nearby mountain in the Five College Depository. This was built by the feds during the Cold War to house bank documents in case of nuclear war, but is now owned by Amherst College. Old journals buried there can be requested, I guess, but the location was more useful during filming of a Mel Gibson thriller a few years ago. –  Jim Humphreys Nov 13 '12 at 23:41
@Jim, great comment! I actually found the Notices here yesterday in the periodicals section of the stacks in the math library at the University of Minnesota and today in the bowels of Rolvaag Library at St. Olaf College. –  Barry Cipra Nov 14 '12 at 0:15
@J.GE, From the abstract, it is not clear what is Calabi's proof, but probably it has the same flavor as this one:… –  Anton Petrunin Aug 18 '13 at 1:15

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