Suppose $f:M\rightarrow N$ is a submersion. Then, $M\times_NM$ is a smooth manifold from Transversal theorem.

Suppose $\theta:G\rightarrow H$ is a morphism of Lie groups. Assume that it is a submersion. Does it imply $G\times_H G$ is a Lie group?

As $G\rightarrow H$ is submersion, $G\times_H G $ is a smooth manifold. Does it have to be Lie group?

When will $G\times_H G$ is a Lie group?


1 Answer 1


Yes it’s always a Lie group.

Closure under multiplication and inverses you can check, so it’s a group. For smoothness, note $M\times_NM$ isn’t just a manifold, but an embedded submanifold of $M\times M$. The multiplication and inversion for $G\times_HG$ are restricted from $G\times G$ to the submanifold, hence smooth.

  • $\begingroup$ Ok. Restriction of smooth map to embeded submanifold is smooth.. So, it i the case.. Thanks $\endgroup$
    – user537667
    Jan 20, 2019 at 5:35
  • $\begingroup$ @PraphullaKoushik No problem! $\endgroup$
    – Ben
    Jan 20, 2019 at 6:00
  • $\begingroup$ I have added more details as a community wiki answer. Let me know if I am missing something. Thank you :) $\endgroup$
    – user537667
    Jan 20, 2019 at 9:21

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