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Metric spaces are isometric if there exists a bijective isometry between them.

Is there a standard notation for this, along the same lines as $X\approx Y$ for homeomorphic spaces and $X\simeq Y$ for homotopy equivalent spaces?

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I would say no. And actually I'm not sure that your notations for homeomorphism and homotopy equivalence, are completely standard... –  Alain Valette Oct 28 '11 at 9:18
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In every case you can write $\cong$ when the category is clear in the context. –  Martin Brandenburg Oct 28 '11 at 9:18
    
@Alain: point taken. I meant to say that the above symbols are standard enough that a topologist would know what they mean without them being defined. I was wondering if there was a similar symbol for isometry ($=$ seems inaccurate, and $\cong$ seems too general). –  Mark Grant Oct 28 '11 at 11:38
    
I don't know of such a notation. But I agree with Martin that if you scrupulously bear in mind the category under discussion, then $\cong$ is all-purpose. If that doesn't seem satisfactory, then I'd lamely offer $\cong_{\text{isom}}$ or something like that for emphasis. –  Todd Trimble Oct 28 '11 at 13:27
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