## Metric Tensor

A 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 between any two points in a given Space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean Theorem (1)

In Euclidean Space, 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 such that the Inner Product is a symmetric, nondegenerate, bilinear form on a Vector Space. This means that it takes two Vectors as arguments and produces a Real Number such that (3) (4) (5) (6) (7)

with equality Iff .

In coordinate Notation (with respect to the basis), (8) (9) (10)

where is the Minkowski Metric. This can also be written (11)

where   (12)   (13) (14)

gives (15)

The metric is Positive Definite, so a metric's Discriminant is Positive. For a metric in 2-space, (16)

The Orthogonality of Contravariant and Covariant metrics stipulated by (17)

for , ..., gives linear equations relating the quantities and . Therefore, if metrics are known, the others can be determined.

In 2-space,   (18)   (19)   (20)

If is symmetric, then   (21)   (22)

In Euclidean Space (and all other symmetric Spaces), (23)

so (24)

The Angle between two parametric curves is given by (25)

so (26)

and (27)

The Line Element can be written (28)

where Einstein Summation has been used. But (29)

so (30)

For Orthogonal coordinate systems, for , and the Line Element becomes (for 3-space)     (31)

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