Riemannian metric

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The metric of a space given by a positive-definite quadratic form. If a local coordinate system $( x ^ {1} \dots x ^ {n} )$ is introduced for the space $V _ {n}$ and if at each point $X( x ^ {1} \dots x ^ {n} ) \in V _ {n}$ functions $g _ {ij} ( X)$, $i, j = 1 \dots n$, $\mathop{\rm det} ( g _ {ij} ) > 0$, $g _ {ij} ( X) = g _ {ji} ( X)$, are defined which are the components of a covariant symmetric tensor of the second valency, then this tensor is called the fundamental metric tensor of $V _ {n}$. The length $ds$ of the covariant vector $( dx ^ {1} \dots dx ^ {n} )$ is expressed using the fundamental tensor:

$$ds ^ {2} = g _ {ij} ( X) dx ^ {i} dx ^ {j} ;$$

the form $g _ {ij} dx ^ {i} dx ^ {j}$ is a positive-definite quadratic form. The metric of $V _ {n}$ determined using the form $ds ^ {2}$ is called Riemannian, and a space with a given Riemannian metric introduced into it is called a Riemannian space. The specification of a Riemannian metric on a differentiable manifold means the specification of a Euclidean structure on the tangent spaces of this manifold depending on the points in a differentiable way.

A Riemannian metric is a generalization of the first fundamental form of a surface in three-dimensional Euclidean space — of the internal metric of the surface. The geometry of the space $V _ {n}$ based on a definite Riemannian metric is called a Riemannian geometry.

There are generalizations of the concept of a Riemannian metric. Thus, a pseudo-Riemannian metric is defined with the aid of a non-definite non-degenerate quadratic form (see Pseudo-Riemannian space and Relativity theory). A degenerate Riemannian metric, that is, a metric form defined with the aid of functions $g _ {ij} ( X)$ for which $\mathop{\rm det} ( g _ {ij} ) = 0$, defines a semi-Riemannian space.

References

 [1] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949) [2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) [3] B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)