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I've been learning a bit about orientability on smooth manifolds. I'm having torubles with this exercise:

Given two smooth manifolds $M$ and $N$, show that the product manifold $M \times N$ is orientable if and only if $M$ and $N$ are orientable.

Using the orientability of $M$ and $N$ one can obtain an oriented atlas for each manifold and the construct an oriented atlas for the product formed by the product charts (i.e. charts of the form $(U \times V, \phi \times \psi)$ with $(U, \phi)$ a chart of $M$ and $(V, \psi)$ a chart of $N$).

I'm stuck on proving the converse. Given an oriented atlas for $M \times N$ one can obtain another oriented atlas formed by charts with basic open sets as domains. But from this atlas I don't know how to extract an atlas for $M$.

I was given the following hint: If $M \times N$ is orientable then $M \times \mathbb{R}^n$ is orientable where $n$ is the dimension of $N$. I don't know either how to prove it nor how to use it.

Pipicito
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2 Answers2

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$(\Rightarrow)$ If $M^m\times N^n$ is orientable, there is a volume form $\eta\in\Omega^{n+m}(M\times N)$. For any fixed point $q\in N$, take a basis $\{w_1,...,w_n\}$ for $T_qN$. Now define $\omega\in\Omega^m(M)$ by: $$\omega(X_1,...,X_m):=\eta_{(\cdot,q)}\left(X_1,...,X_m,w_1,...,w_n\right)$$

We prove that $\omega$ is a volume form. For a fixed $p\in M$, take a basis $\{v_1,...,v_m\}$ for $T_pM$. Using the identification $T_{(p,q)}M\times N\equiv T_pM\oplus T_qN$, then $\{v_1,...,v_m,w_1,...,w_n\}$ is a basis for $T_{(p,q)}M\times N$. Since $\eta$ is a volume form, $\omega_p(v_1,...,v_m)=\eta_{(p,q)}(v_1,...,v_m,w_1,...,w_n)\neq 0$, which means that $\omega$ is a volume form, so $M$ is orientable. By a similar argument, $N$ is orientable.$_\blacksquare$

$(\Leftarrow)$ If $M,N$ are orientable, there are volume forms $\omega\in\Omega^m(M), \sigma\in\Omega^n(N)$. For the natural projections $\pi_M:M\times N\to M$ and $\pi_N:M\times N\to N$, define: $$\eta:=\pi_M^*(\omega)\wedge \pi_N^*(\sigma)\in\Omega^{n+m}(M\times N)$$

We prove that $\eta$ is a volume form. For a fixed $x=(p,q)\in M\times N$, take basis $\{v_1,...,v_n\}$ for $T_pM$ and $\{w_1,...,w_n\}$ for $T_qN$. Because $\omega$ and $\sigma$ are volume forms, we have $\omega_p(v_1,...,v_m)\neq 0$ and $\sigma_q(w_1,...,w_n)\neq 0$. Since $\{v_1,...,v_m,w_1,...,w_n\}$ is a basis for $T_x(M\times N)$, it's enough to check that $\eta_x(v_1,...,v_m,w_1,...,w_n)\neq 0$. Indeed, noticing that $(d\pi_M)_x(v_i)=v_i$, $(d\pi_M)_x(w_j)=0$, $(d\pi_N)_x(v_i)=0$ and $(d\pi_N)_x(w_j)=w_j$, we have: $$\eta_x(v_1,...,v_m,w_1,...,w_n)=\underbrace{\omega_p(v_1,...,v_m)}_{\neq 0}\,\underbrace{\sigma_q(w_1,...,w_n)}_{\neq 0}\neq 0\,\,_\blacksquare$$

rmdmc89
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    I think it's wrong at the beginning of "Define $\eta\in\Omega^{n+m}(M)$" must be "Define $\eta\in\Omega^{n+m}(M\times N)$". And even then, I think the product is not well defined. In $\Lambda ^{m+n}(M\times N)$ has more vectors than $\omega \sigma$ can actually receive. Am I right? Think of the case $m=n=1$ you will see. I have a problem similar to posted, my idea was to do as you but I found this problem. How to correct? – Mancala Jun 10 '18 at 16:35
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    Indeed, $\eta\in\Omega^{n+m}(M\times N)$ is defined by $\eta = \pi_1^\omega\wedge\pi_2^\sigma$. But your definition is wrong, as vector fields on $M\times N$ needn't necessarily come as vector fields on $M$ or vector fields on $N$. You really need to project vector fields down to the respective factors and you need to use the complete skew symmetry of the wedge product. – Ted Shifrin Jun 10 '18 at 20:45
  • @TedShifrin, you're absolutely right. I've tried to develop your idea, hope it's correct. – rmdmc89 Jun 11 '18 at 00:21
  • @Mancala, I've followed TedShifrin's suggestion. Thanks for noticing the mistake – rmdmc89 Jun 11 '18 at 00:21
  • (⇒) How do we know whether $\omega$ is smooth? $w_i$ seem not to be fixed vectors, but rather their images under identifications $T_{(p,q)}M\times N \simeq T_pM \oplus T_qN$ (and the map is different at each point). – Paweł Czyż Nov 10 '19 at 16:52
  • @PawełCzyż, it's a fair question. I implicitly used the fact that the identifications $T_{(p,q)}(M\times N)\simeq T_pM\times T_qN$ actually characterize a vector bundle isomorphism $T(M\times N)\simeq \pi_1^(TM)\oplus \pi_2^(TN)$, where the $\pi_1:M\times N\to M$ and $\pi_2:M\times N\to N$ are the natural projections and $\pi_1^(TM),\pi_2^(TN)$ are the pullback bundles. Intuitively this means that if $p,q$ vary smoothly, the map $T_{(p,q)}(M\times N)\to T_pM\times T_qN$ varies smoothly, which is what you're asking. – rmdmc89 Nov 10 '19 at 22:03
  • @PawełCzyż, for more details, see this https://math.stackexchange.com/questions/1337476/tangent-bundle-of-product-manifold – rmdmc89 Nov 10 '19 at 22:03
  • @rmdmc89 Thanks – I have been thinking about defining a vector field that does the job explicitly, but suggested link offers a much more elegant solution. – Paweł Czyż Nov 12 '19 at 17:19
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A manifold is orientable iff it admits a volume form (a nowhere vanishing top degree alternating differential form). It follows that an open submanifold of an orientable manifold is orientable. Take an open subset $U\subset N$ diffeomorphic to $\mathbb R^n$. Then $M\times U$ is an open submanfold of $M\times N$, hence orientable. Now an orientation on $U$ and $M\times U$ defines an orientation on $M$.

Gil Bor
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