Category:Metric Tensors

This category contains results about Metric Tensors.
Definitions specific to this category can be found in Definitions/Metric Tensors.

Consider a smooth manifold $\MM$ on the real space $\R^n$.

Consider a Riemannian metric $\d s$ on $\MM$ between nearby points $\tuple {x_1, x_2, \ldots, x_n}$ and $\tuple {x_1 + \d x_1, x_2 + \d x_2, \ldots, x_n + \d x_n}$ defined in the form:

$\ds \d s^2 = \sum_{i, j \mathop = 1}^n \map {g_{i j} } x \rd x_i \rd x_j$

where each $g_{i j}$ is a suitable real-valued function of $x_1, \ldots, x_n$.

The $\map {g_{i j} } x$ are the components of a symmetric covariant tensor field.

This symmetric covariant tensor field is known as the metric tensor on $\MM$.