A separable measure space has a natural quotient set can be properly metrized by the induced metric.
3.
There remains ambiguity in the action up to an arbitrary functional of the induced metric h _ { ab }.
4.
Because the induced metric is fixed on \ partial \ mathcal { M }, the only quantity to be varied is K is the trace of the extrinsic curvature.
5.
The precise form of the Hartle Hawking state is the path integral over all " D "-dimensional geometries that have the required induced metric on their boundary.
6.
It is the Hausdorff distance between the unit balls of the two C *-algebras, under the norm-induced metric on the space of all bounded operators on that Hilbert space.
7.
In mathematics, a "'polar action "'is a proper and flat with respect to the induced metric, then the action is called "'hyperpolar " '.
8.
A point " p " in a Riemannian submanifold is umbilical if, at " p ", the ( vector-valued ) Second fundamental form is some normal vector tensor the induced metric ( First fundamental form ).
9.
Although the induced metric of g _ 1 and g _ 2 will agree on \ Sigma _ 2, the normal derivative of g _ 1 at \ Sigma _ 2 will not in general be equal to that of g _ 2 at \ Sigma _ 2.
10.
X ^ \ mu ( \ xi ) is the field on the worldsheet describing the embedding of the string in 25 + 1 spacetime; in the Polyakov formulation, g is not to be understood as the induced metric from the embedding, but as an independent dynamical field.
