Physical differences between our universe and strongly negatively curved one Interestingly, the idea of a negatively curved universe isn't entirely science fiction. In particular if decrease a single physical constant (known as the density parameter) to be less than one, the universe as we know will be negatively curved. This is nice because for a story in which you want a negatively curved universe, you don't need to invent a new set of laws of physics; you can just adjust the density parameter to be less than one, and then use normal physical laws.
My question is then, just how different would the universe be if it where negatively curved. In particular, let's say the universe had a curvature of -100 per mi squared (-40 per km squared). What phenomena would be significantly different in that case?
In particular, I am wondering if human like life could still form in such a universe.
EDIT: It appears that electromagnetism decays exponentially, for one.
 A: One thing to note about relativistic cosmology is its solutions evolve in time. So if at one moment the universe has a specific value of spatial curvature the next moment it would be different. The value of curvature  specified in OP is quite large and thus it correspond to just a specific moment near the Big Bang (or Big Crunch) of this hypothetical universe.
Another thing is that providing just the curvature and openness is not enough to specify the evolution of the universe even with the additional constraint that we "don't need to invent a new set of laws of physics". 
Evolution of the universe on large scales could be a modeled by a solution of Friedmann equations. By specifying openness we assume $k=-1$ and spatial curvature specifies the scale factor at that moment $a(t_0)$, which for the OP value would be on the order of a hundred meters. One also has to specify time derivative $\dot{a}$ and matter content, which enters the equations through energy $\rho$ and pressure $p$. 
Let us have a look at the first Friedmann equation:
$$ \frac{\dot{a}^2 + kc^2}{a^2} = \frac{8 \pi G \rho + \Lambda c^2}{3} $$
The right hand side should be positive, if one assumes that the matter is something like what we have in our universe (i.e. there are no negative energies, including dark one). We see than that this immediately puts a constraint on a speed of expansion: $|\dot{a}|>c$ at all times. This means that all matter that is accessible to observation (including future observations) is located within volume of a sphere with radius smaller than $a$ (Hubble volume). In particular,  exponential decay of electrostatic force is now replaced by more complicated phenomenon: as soon as the distance between charges becomes comparable to $a$, one has to include non-stationarity of spacetime.
We see, that specifying $\dot{a}$ also gives us the energy density. For example if we specify $\dot{a}=2 c$, when $a=200\,\text{m}$, then the energy density would be about $10^{22}\,\text{kg}/\text{m}^3$. This is large, of course, but inside the Hubble volume there would be only a fraction of the solar mass of matter. And this matter would rapidly expand as it cools off without forming a single gravitationally bound object: no stars, no planets, no comets, no complex chemistry.
To combat this uninteresting fate we could try to increase $\dot{a}$ (while keeping the value of $a$ constant). This increases energy density in the r.h.s. but also decreases overall role of term with $k$ (at least during early stages of expansion). At the same time this pushes the moment at which a given value of $a$ achieved closer and closer to the Big Bang singularity. And if there is enough matter in the Hubble volume, there could be formation of stars, star clusters, and galaxies just like in our universe. Within such gravitationally bound systems the effects of openness of the universe would be negligible, however observer  that evolved in such a world would notice far fewer or even completely absent large scale structures such as galactic groups, clusters, superclusters and that the universe is expanding more rapidly than ours.
Now, I understand that OP's desire is to have a universe that is spatially hyperbolic yet almost static. I do not know of the way one could achieve this in a somewhat realistic cosmological model, however such universe could in principle be constructed with the help of negative cosmological constant $\Lambda$. Such solution would be an extension of static Einstein universe model on the case of negative spatial curvature (and negative cosmological constant). This requires precise matching between the values of $a$, $\Lambda$ and $\rho$ and the solution would unstable against small perturbations.
A: If you mean a contracting universe?
If the universe was contracting we would be bombarded with ever increasingly blue shifted radiation.
As light is blue shifted the wavelength decreases, at some point it would go through x-ray and even gamma-ray frequencies.
I'm pretty sure being bathed in gamma rays would be bad, for any form of life.
How much negative curvature it would take I am not sure, but -100 per mi squared seems quite high. 
