I have a pair of smooth 3-vector fields $B: \mathbb R^3 \to \mathbb R^3$ and $J: \mathbb R^3 \to \mathbb R^3$, with $\nabla \times B = J$, both of which are tangential to a regular surface defined as P = constant for some $P: \mathbb R^3 \to \mathbb R$(i.e $ B \cdot \nabla P = 0 = J \cdot \nabla P $ ). Both the vector fields do not vanish anywhere on the surface. Additionally, $\nabla \cdot B = 0 $, which means both fields are divergence free.
Assuming said surface is compact and boundary-less, can we conclude - it is homeomorphic to either a torus or a Klein bottle? (Of course the Klein bottle can be dismissed as we are talking of surfaces embedded in $\mathbb R^3$.)
Does any one know where can I find the English translation of Alexandroff and Hopf's Topologie 1935e? The article in which this came up is an old plasma physics paper (Kruskal-Kulsrud 1958) which refer to Satz III in page 552 of this book, so I'd be thankful if can just name the theorem I am looking for and give a reference.
I could read off some key words and understand that this has to do with the signed intersection numbers at fixed points summing to the Euler characteristics of the surface (this theorem I have read elsewhere, so I guess this is the one in the Topologie book ), but what is its exact statement and how to apply this theorem in this case?