Gravity Warps Space & Time: A cm Difference?

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If gravity warps space as well as time, does that mean that a cm in a high gravity area would be smaller than a cm in a low gravity area?
 
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To you, a cm would still be a cm. For example, if you'd put your ruler next to an object in a high gravity area, the ruler would also be warped and you'd still get the same measurement. What you're actually measuring is the proper length (I think that mathematically, it just occurred to me, you're using Riemann Normal Coordinates, in which the space is locally flat). To an outside observer, however, the object would look differently, because the metric at that point is different.
 

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