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Not an original source, but the Wikipedia page [0] gives a good introduction. A contact manifold is then just a manifold equipped with a contact structure.

A more in-depth discussion can be found in many differential topology texts, e.g. page 581 of [1].

[0] https://en.m.wikipedia.org/wiki/Contact_geometry

[1] Lee, Introduction to Smooth Manifolds https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...

Way outside of my field - but from the link above: - https://www.jstor.org/stable/1970165 On Contact Manifolds Their reference - https://core.ac.uk/reader/82059800 - A Brief History of Contact Geometry and Topology - HansjSrg Geiges

Seems solid enough. There's lots of other fun links there. I'm fascinated by this field but I don't have a degree of any kind so ... lots more learning to go!!!

but when I understand manifolds both contermporary flat-earthers and scientific academics(?) get mad at me?

but I wouldn't claim to understand manifolds, I just understand that they have something to do with flatness perception at small scales (or short distances)