| 1. | This cannot be done with the ordinary Euclidean metric.
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| 2. | The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics.
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| 3. | Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric.
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| 4. | The Minkowski metric is not a Euclidean metric, because it is indefinite ( see metric signature ).
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| 5. | For parallel transport on a sphere, the implicit dependence is intrinsic to the non-euclidean metric.
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| 6. | The Euclidean metric defines the distance between two points as the length of the straight line connecting them.
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| 7. | The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.
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| 8. | Remember, to measure all distances, I am using Euclidean metric, NOT the taxi-cab metric.
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| 9. | The seventh cross product is for the Pauli spin in the 3-dimensional space with the Euclidean metric.
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| 10. | This embedding induces a metric on the sphere, it is inherited directly from the Euclidean metric on the ambient space.
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