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"Are there simple properties that

You are necessary and sufficient?"asking for conditions on a metric space $X$ for which any distance preserving map $X\to X$ is bijective. (Usually isometry is defined as bijective distance preserving map).

Well, there are arbitrarily bad spaces $X$ such that the only distance preserving map $X\to X$ is identity. In this case distance preserving maps (well, the only one) form the trivial group.

So, I do not see a language which could be used to formulate such a necessary and sufficient condition. For sufficient conditions:

• Compactness;
• Proper + cocompact isometric group action. (proper=bounded closed sets are compact)
• Any complete connected space for which the domain invariance theorem holds; in particular complete connected Riemannian manifolds without boundary.

"Are there simple properties that are necessary and sufficient?"

Well, there are arbitrary arbitrarily bad spaces $X$ such that the only distance preserving map $X\to X$ is identity. In this case distance preserving maps (well, the only one) form the trivial group.

So, I do not see a language which could be used to formulate such a necessary and sufficient condition. For sufficient conditions:

• Compactness;
• Proper + cocompact isometric group action. (proper=bounded closed sets are compact)
• Any complete connected space for which the domain invariance theorem holds; in particular complete connected Riemannian manifolds without boundary.
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"Are there simple properties that are necessary and sufficient?"

Well, there are arbitrary bad spaces $X$ such that the only distance preserving map $X\to X$ is identity. In distance preserving maps (well, only one) form the trivial group.

So, I do not see a language which could be used to formulate such a necessary and sufficient condition. For sufficient conditions:

• Compactness;
• Proper + cocompact isometric group action. (proper=bounded closed sets are compact)
• Any space for which the domain invariance theorem holds.