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What is the a connection between isometry groups and Lie manifolds? A Lie group is a...

What is the a connection between isometry groups and Lie manifolds?

A Lie group is a smooth manifold that is also a group. e.g. U(1) is the circle, SU(2) is the 3-sphere , SO(3) is the real-projective-3-space. A manifold can also have an isometry group e.g. the isometry group of a 2-sphere is O(3). Is this correct? Please explain your answer.

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What is the a connection between isometry groups and Lie manifolds?
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