The MetricTensor generalises the notion of "distance" into arbitrarily many dimensions, and in possibly curved spaces. The MetricTensor doesn't '''measure''' distance, it '''defines''' distance. 

Euclidean spaces are "flat" because their metric tensors define distance such that Pythagoras' Theorem is true. Non-euclidian spaces have MT's that define the  square of the hypotenuse to be different from the sum of the squares of the other two sides (or however many in your dimensionality).

Masses act to locally distort the metric tensor. Light always moves along paths that mimimise certain quantities wrt that tensor. voila: gravitational lensing. TheEarthIsFlatButSpaceIsCurved.

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CategoryMath