| 释义 |
Metric TensorA Tensor, also called a Riemannian Metric, which is symmetric and Positive Definite. Very roughly, the metric tensor  is a function which tells how to compute the distance betweenany two points in a given Space. Its components can be viewed as multiplication factors which must be placed infront of the differential displacements  in a generalized Pythagorean Theorem
 | (1) |
 where  is the Kronecker Delta (which is 0 for  and 1 for  ), reproducing the usual form of the Pythagorean Theorem
 | (2) |
The metric tensor is defined abstractly as an Inner Product of every Tangent Space of a Manifold suchthat the Inner Product is a symmetric, nondegenerate, bilinear form on a Vector Space. This means that ittakes two Vectors  as arguments and produces a Real Number  such that
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 .
In coordinate Notation (with respect to the basis),
 | (8) |
 | (9) |
 | (10) |
 is the Minkowski Metric. This can also be written
 | (11) |
 | (14) |
 | (15) |
 | (16) |
 | (17) |
 , ...,  gives linear equations relating the  quantities and  . Therefore, if metrics are known, the others can be determined.
In 2-space,
If  is symmetric, then
In Euclidean Space (and all other symmetric Spaces),
 | (23) |
 | (24) |
 between two parametric curves is given by
 | (25) |
 | (26) |
 | (27) |
The Line Element can be written
 | (28) |
 | (29) |
 | (30) |
 for , and the Line Element becomes (for 3-space)
where  are called the Scale Factors.See also Curvilinear Coordinates, Discriminant (Metric), Lichnerowicz Conditions, Line Element,Metric, Metric Equivalence Problem, Minkowski Space, Scale Factor, Space
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