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A Riemannian metric is a metric with a positive definite signature (v, 0). A Lorentzian metric is a metric with signature ( p , 1) , or (1, p ) . There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as ( v − p ) , where v and p are as above, which is equivalent to the above definition when ...
Equivalently, the metric has signature (p, n − p) if the matrix g ij of the metric has p positive and n − p negative eigenvalues. Certain metric signatures which arise frequently in applications are: If g has signature (n, 0), then g is a Riemannian metric, and M is called a Riemannian manifold.
A pseudo-Riemannian manifold (M, g) is a differentiable manifold M that is equipped with an everywhere non-degenerate, smooth, symmetric metric tensor g. Such a metric is called a pseudo-Riemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.
A smooth manifold endowed with a Riemannian metric is a Riemannian manifold, denoted (,). [3] A Riemannian metric is a special case of a metric tensor. A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally, a constant pseudo-Riemannian metric in Cartesian coordinates. As such, it is a nondegenerate symmetric bilinear form, a type (0, 2) tensor.
A representative Riemannian metric on the sphere is a metric which is proportional to the standard sphere metric. This gives a realization of the sphere as a conformal manifold. The standard sphere metric is the restriction of the Euclidean metric on R n+1 = + + + +
Cartan connection. Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.. Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete ...