Orientation of a product manifold and the induced orientation on the tangent spaces

For oriented manifolds $$M$$ and $$N$$, I want to give $$M\times N$$ an orientation. (*) Additionally, I want to show that this orientation makes positively oriented bases $$(v_1, \dots , v_m)$$ and $$(w_1, \dots , w_n)$$ for $$T_pM$$ and $$T_qN$$ give a positively oriented basis $$(v_1 \times 0, \dots , v_m \times 0, 0 \times w_1, \dots , 0 \times w_n)$$ for $$T_{(p,q)}(M \times N) = T_pM \times T_qN$$.

Let $$\{(U_i,\varphi_i)\}_{i\in I}$$ be an oriented atlas for $$M$$ and $$\{(V_j,\psi_j)\}_{j\in J}$$ an orientable atlas for $$N$$. Then $$M\times N$$ has product atlas generated by the charts $$\{(U_i\times V_j,\varphi_i\times \psi_j)\}_{i\in I,j\in J}$$. These charts cover $$M\times N$$. Since the charts on $$M$$ and $$N$$ are orientable, it follows that for each $$p\in U_i\cap U_j$$ and $$q\in V_i\cap V_j$$ that the Jacobian matrices $$d(\varphi_j\circ\varphi_i^{-1})_p$$ and $$d(\psi_j\circ \psi_i^{-1})_q$$ have positive determinant.

But then $$\det(d((\varphi_j\times\psi_j)\circ (\varphi_i\times \psi_i)^{-1})_{(p,q)})=\det(d(\varphi_j\circ\varphi_i^{-1})_p\times d(\psi_j\circ\psi_i^{-1})_{q})=\det(d(\varphi_j\circ\varphi_i^{-1})_p)\det(d(\psi_j\circ\psi_i^{-1})_q)>0$$

What I don't know is how an oriented manifold (defined in terms of an oriented atlas, that is an atlas generated by a choice of covering charts, where all chart transitions have positive determinant jacobian matrices) induces an orientation on each tangent space.

Do we just fix an oriented basis $$\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in I}$$ for the $$n$$-dimensional manifold $$M$$, and for each $$p\in M$$ where $$U_\alpha\ni p$$ take $$\varphi_\alpha^*\omega$$ for $$\omega$$ the standard orientation form on $$\Bbb R^n$$, which is well defined since chart transitions about $$p$$ preserve orientation.

• Given oriented charts, you can find a positive basis at a point by pulling back the standard basis from $\Bbb{R}^n$ via the chart. Since you've defined your charts on the product to be $\phi_i\times \psi_j$, this makes it immediately clear that $(*)$ holds. – jgon Jun 14 at 23:15
• @jgon Sorry, I don't follow. You mean for an $m$-dimensional manifold $M$ with a chosen oriented atlas (in the sense that the derivative of the chart transitions all have positive determinant), how do I give $T_pM$ an orientation? What's meant by pulling back the standard basis from $\Bbb R^n$? Do you mean we let $(U,\varphi_U)$ be our oriented chart, then let $det$ be the standard orientation form on $\Bbb R^m$ and just pull this back over $\varphi_U$? – F.White Jun 19 at 15:59
• @jgon Maybe I did follow by the end of posting that :P. Is my last paragraph in the edit what you mean? – F.White Jun 19 at 16:09
• So your question is not really about the product of manifolds, but you want to understand the connection between an oriented atlas and orientations of the tangent spaces? – Paul Frost Jul 4 at 16:13