A description/proof of that intersection of 2 transversal regular submanifolds of \(C^\infty\) manifold is regular submanifold of specific codimension
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold.
- The reader knows a definition of regular submanifold.
- The reader knows a definition of transversal regular submanifolds.
- The reader admits the transversality theorem: for any transversal map, the preimage of the regular submanifold on the codomain is a regular submanifold of codimension of the codimension of the regular submanifold on the codomain.
- The reader admits the proposition that any regular submanifold of any regular submanifold of any \(C^\infty\) manifold is a regular submanifold of the base manifold, of codimension of the codimension of the child submanifold plus the codimension of the grandchild submanifold with respect to the child submanifold.
Target Context
- The reader will have a description and a proof of the proposition that the intersection of any 2 transversal regular submanifolds of any \(C^\infty\) manifold is a regular submanifold of codimension of the sum of the codimensions of the regular submanifolds.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any \(C^\infty\) manifold, M, and any transversal regular submanifolds, \(M_1, M_2 \subseteq M\), the intersection of the transversal regular submanifolds, \(M_1 \cap M_2\), is a regular submanifold of \(M\) of codimension of the sum of the codimensions of \(M_1\) and \(M_2\).
2: Proof
Think of the inclusion map, \(i: M_1 \rightarrow M\), which is \(C^\infty\) because \(M_1\) has the adapting charts and \(i\) with respect to any adapting chart and the adapted chart is obviously \(C^\infty\). \(i\) is a transversal map from \(M_1\) to \(M_2\), because for each \(p \in i^{-1} (M_2)\), \(i_* (T_pM_1) + i'_* (T_{i (p)}M_2) = T_{i (p)}M\) where \(i'\) is the inclusion map, \(i': M_2 \rightarrow M\), because as \(i\) is an inclusion, \(i (p) = p\), so, that equation equals \(i_* (T_pM_1) + i'_* (T_pM_2) = T_pM\), which is nothing but the definition of the transversality of \(M_1\) and \(M_2\) (although the definition is prevalently expressed like "\(T_pM_1 + T_pM_2 = T_pM\)", it is really a sloppy expression because any vector on \(T_pM_1\) cannot be added to any vector on \(T_pM_2\) because they are on different spaces; the vectors can be added only because they are pushed forward to the same \(T_pM\) space). By the transversality theorem: for any transversal map, the preimage of the regular submanifold on the codomain is a regular submanifold of codimension of the codimension of the regular submanifold on the codomain, \(i^{-1} (M_2) = M_1 \cap M_2\) is a regular submanifold of \(M_1\) of codimension of the codimension of \(M_2\) (with respect to \(M\)), denoted as \(codim (M_2, M)\). By the proposition that any regular submanifold of any regular submanifold of any \(C^\infty\) manifold is a regular submanifold of the base manifold, of codimension of the codimension of the child submanifold plus the codimension of the grandchild submanifold with respect to the child submanifold, \(M_1 \cap M_2\) is a regular submanifold of \(M\) of codimension of \(codim (M1, M) + codim (M2, M)\).
