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Let's suppose that I have a Lie group G parametrized by one real scalartand acting on ℝ^{2}.

Is it generally correct to say that theorbitsof the points of ℝ^{2}under the group action are one-dimensional submanifolds of ℝ^{2}, because G is parametrized by one single scalar?

If so, how can I prove this statement?

Thanks.

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# Lie group actions and submanifolds

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