[Tainnor]

> At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done.

I'm not sure. I only have a rather rudimentary understanding of topology, so I do understand the definition of a manifold on a technical level, but I don't know any interesting examples or theorems about them so it wouldn't be immediately clear to me why something being a submanifold is worth mentioning.

Similarly, I don't think that just reading the definition really gives you a good understanding of groups. You probably want to work through some examples of groups, and arguably, the importance of groups doesn't really become clear until you've encountered group actions.

[fpgamlirfanboy]

You skipped over the second sentence of what you're responding to:

> Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with

In such a case you're not interested in either manifold or sub-manifold or group in and of itself. So a lack of familiarity with theorems isn't an impediment.

[Tainnor]

I didn't "skip over" anything. If it's irrelevant to you that something is a submanifold or group, you also don't need to look up the definition.

Reading mathematical definitions on their own just doesn't give you a whole lot of context about the objects they're describing.