If ? is a doubly connected planar region, then there is a diffeomorphism " F " of an annulus " r " d " | z | d " 1 onto the closure of ?, such that after a conformal change the induced metric on the annulus can be continued smoothly by reflection in both boundaries.
12.
A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds . ( The same goes for a more general concept, holomorphic bisectional curvature . ) For example, every complex submanifold of "'C " "'n " ( with the induced metric from "'C " "'n " ) has holomorphic sectional curvature d " 0.
13.
Here we utilized the cross-normalization condition l ^ an _ a = n ^ al _ a =-1 as well as the requirement that g _ { ab } + l _ an _ b + n _ al _ b should span the induced metric h _ { AB } for cross-sections of { v = constant, r = constant }, where dv and dr are not mutually orthogonal.
14.
In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, q _ { ab } ( x ), on the spatial slice ( the metric induced on the spatial slice by the spacetime metric ), and its conjugate momentum variable related to the extrinsic curvature, K ^ { ab } ( x ), ( this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time ).
15.
In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, q _ { ab } ( x ), on the spatial slice ( the metric induced on the spatial slice by the spacetime metric ), and its conjugate momentum variable related to the extrinsic curvature, K ^ { ab } ( x ), ( this tells us how the spatial slice curves with respect to spacetime and is a measure of how the induced metric evolves in time ).
16.
Where \ mathcal { S } _ \ mathrm { EH } is the Einstein Hilbert action, \ mathcal { S } _ \ mathrm { GHY } is the Gibbons Hawking York boundary term, h _ { ab } is the induced metric ( see section below on definitions ) on the boundary, h its determinant, K is the trace of the second fundamental form, \ epsilon is equal to + 1 where \ partial \ mathcal { M } is timelike and-1 where \ partial \ mathcal { M } is spacelike, and y ^ a are the coordinates on the boundary.
