Perhaps the question is self-explanatory. The context is Kleiner's Inv. Math. paper An isoperimetric comparison theorem, where the statement of the main theorem begins with "Let $M$ be a complete one-connected three-dimensional Riemannian manifold...".

The article does not define what one-connectedness is, and google is not much help either. The article makes reference to books by Gromov(-Lafontaine-Pansu) and Burago-Zalgaller, and a JDG article by Aubin, where maybe this is defined, but none of these references are available to me now.

(I know that we say that a subspace $A\subset X$ is $k$-connected if the relative homotopy groups $\pi_\ell(X,A)$ vanish for $\ell\leq k$, but here there doesn't seem to be a distinguished subspace.)

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    $\begingroup$ I have seen "1-connected" as an abbreviation for "simply connected". $\endgroup$ – Zhen Lin Nov 25 '13 at 8:23
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    $\begingroup$ Seems that $k$-connected means $\pi_i (M) = 0$ for all $i\leq k$. $\endgroup$ – user99914 Nov 25 '13 at 8:38

As mentioned in the comments, we say that a (non-empty) space $X$ is $k$-connected if $\pi_i(X)$ is trivial for all $i\leq k$. In particular, if $X$ is $0$-connected, then this just says that $X$ is path-connected, and if $X$ is $1$-connected, then this just says that $X$ is simply connected (which a priori implies $X$ is path-connected).

For general $k$-connected spaces, this becomes the right setting for applying the Hurewicz theorem which relates the first non-trivial homotopy group with homology in the same degree in a simple way (as well as giving further information about the Hurewicz map in higher degrees).


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