Sorry, but as soon as you do any normailzation at all, you ARE changing the distribution.
It may be that you are not changing the distribution in any way that is significant to you, but it is somewhat difficult to know what you would find significant.
As well, you have not even said what normalization means to you, since I can certainly think of different ways you might attempt such an act, some of them more disruptive than others to the "distribution" of the numbers.
So, consider an arbitrary set of numbers, living in R^3. You can visualize this set as describing a sort of 3-dimensional potato, floating in space.
Simplest might be a simple shift, or translation. Can we shift the entire potato anywhere in space, by adding a constant to all numbers? Such a translation is just a shift of the implicit origin of that space, but it does indeed change the "distribution" in at least a simple sense. We might add the same number to each of x, y, and z, or a different value to each. But if we add a different value to each channel, this might be argued to be a more significant distortion to the distribution. How do I know what you would take to be significant?
Next, we might scale each of x, y, and z by some constant. The same constant, or one that is specific to each channel. Does that matter? Well, it does, again, to some extent. Suppose I start with a set of points on a perfect sphere, but then scale each of x, y, and z by different constants? Now, what was once a perfect sphere is now an ellipsoid, so no longer a sphere. So while we can translate a sphere around anywhere we want by moving the implicit origin, after that translation, a sphere is still a sphere, is still a sphere. However, an ellipsoid is NOT a sphere. Topologically, the two are equivalent though, in the sense that no holes or strange folds were introduced.
So it really may matter what you consider to be validly normalized. When has the distribution changed?
Consider a simple example, in only ONE dimension. Since you talk about distributions, I'll do exactly that.
Here, I have created a random variable X. X is in fact lognormally distributed. But now, let me perform asimple translation on X.
Y is a translated version of X. And, while X was indeed lognormally distributed, Y does NOT follow a lognormal distribution. (You could describe it as a shifted lognormal distribution though.)
I know, this may all seem pedantic. But the fact is, you said you did not wish to change the distribution, and any normalization will indeed do exactly that, on at least some level.
