Suppose $S_1, S_2$ are two orientable surfaces. I am wondering if I can find all splittings of the $T(S_1\times S_2)$ into two $2$-dimensional subbundles.
Up to probably the $4$-sheeted covering, it is enough to consider the splittings into two orientable subbundles. Also, because $S_1\times S_2$ is a complex manifold, and the first Chern class completely classifies complex line bundles, I think I would have the following. If I denote the splitting as $\xi\oplus\eta$, then $$(1+c_1(\xi))(1+c_1(\eta))=(1+c_1(TS_1))(1+c_1(TS_2)).$$ where $c_1(\xi),c_1(\eta)\in H^2(S_1\times S_2)$, $c_1(TS_1)\in H^2(S_1)$, and $c_1(TS_2)\in H^2(S_2)$.
Simplifying this, and taking into consideration the degrees, I have $c_1(\xi)+c_1(\eta)=c_1(TS_1)+c_1(TS_2)$ and $c_1(\xi)c_1(\eta)=c_1(TS_1)c_1(TS_2)$.
Now, should $c_1(\xi)$ and $c_1(\eta)$ just be the solutions to $X^2-(c_1(TS_1)+c_1(TS_2))X+c_1(TS_1)c_2(TS_2)=0$ where $X$ has degree $2$? If it is, is this thing solvable?
I also noticed $X$ can have some crossed terms in $H^1(S_1)\otimes H^1(S_2)$, by the Künneth formula, so I am guessing it would just depend on two integers in $H^2(S_1)$ and $H^2(S_2)$, and how the cross terms would be?
Any thoughts would be appreciated!