Is there a meaningful way to transform logical equations (for instance $a \implies b$; $b \land a \implies c$ etc.) into geometrical representation in spaces such $\mathbb R^n$, $\mathbb C^n$ or manifolds?


I am curious if there is a way to analyze logic equations via transforming them into geometrical objects. Sorry for being too ambiguous, I have no clue where to start looking at. Thank you.



Representing logic formulae in manifolds
One possible interpretation of your somewhat vague question is discussed in Steve Vickers' book 'Topology via Logic'. The interpretation is more on the level of open sets than `big things' like manifolds, but the discussion in the early parts of the book may be useful for you to help you reformulate your question at a slightly more precise level.


Another point worth making is that there are MANY different logics. Have a look at Wikipedia on modal logics and in particular on S4 and S5. Again these operate on the level of the open sets, but some topological constructs lead to interesting logical consequences. The key idea is that of topological semantics of various logics.


A final area nearer to your manifolds etc. would be in interpretations of quantum logic in terms of cobordisms and other similar things. Here key ideas are more likely to be found in theoretical computer science papers or coming from theoretical physics. The ideas in both operad theory and quantum field theories are also worth glancing at but beware there are perhaps too very many directions in which that can lead you!


