I strongly recommend that you share your paper, not only with your friends and family but also with the entire academic community. Upload your paper as a preprint to arXiv. This is the major way for papers to be shared before publication (and after), so that other researchers can learn about your results without waiting for the peer review process (which in mathematics is particularly slow).
(Of course, you'll need to make sure your co-authors agree before you do this.)
See Why upload to academic preprint sites like arXiv?
You may want to double-check that your intended journal will still publish papers when their preprints are on arXiv, but I've never heard of any reputable publisher in mathematics having a problem with this.
If you share a paper before it has been submitted, it is theoretically possible that other people could steal your work, etc, before you submit it. What about after you have submitted your paper?
I would not worry about it. The journal has a record that you submitted it, which would be powerful evidence in case of a priority dispute. Posting to arXiv is an even stronger form of evidence, since it's automatically visible to the public.
In most cases, "stealing work" is a minimal risk, compared to the much larger risk of not getting sufficient attention for your work. It's usually best to err to the side of sharing.