Usually, manifolds have certain internal properties which are being studied on their own. However, I was wondering if a field of mathematics exists where several separate manifolds are considered together (for example overlapping) and influence the internal properties of each other? If this field of mathematics exists, please let me know how it is called and if you know some good textbook on it. Thankful for any directions!
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I would say that your first sentence is not an accurate description of the study of manifolds; perhaps that is only what you have encountered so far?
The field of differential topology is all about manifolds (internally) and the relations (external) between them. The concept of cobordism, for example, is an equivalence relation amongst all closed, connected manifolds: two manifolds $M,N$ of the same dimension are cobordant if there exists a manifold $W$ of one dimension higher such that the boundary of $W$ is diffeomorphic to the disjoint union of $M$ and $N$. Differential topology studies cobordism (and related equivalence relations amongs manifolds) to great depth.
You might look at the classic books of John Milnor, such as "Topology from the differentiable viewpoint", "Characteristic classes", and "Lectures on the h-cobordism theorem", and the recent textbooks of Allen Hatcher such as "K-theory".
Lee Mosher
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You are right! It seems, I simply did not encounter a case where manifolds interact until now. The concept of cobordism sounds interesting. I will definitely look into the suggested literature. Also, I wonder if there is any concept of global dynamical interaction of several manifolds, preferably without imbedding them in any higher dimensional one? – Kagaratsch May 11 '14 at 22:23