A note on an isometric imbedding of upper half-space into the anti de Sitter space
The paper deals with the upper half-space \(U_ n=\{(x_ 1,...,x_ n)\in R^ n| x_ n>0\}\) with the Lorentz metric \(ds^ 2=(-dx^ 2_ 1+dx^ 2_ 2+...+dx^ 2_ n)/x^ 2_ n\). \(U_ n\) is proved to be not geodesically complete. Moreover the isometry group of \(U_ n\) is studied. Furthermore, let \(H^ n_ 1=\{(u_ 0,u_ 1,...,u_ n)\in R^{n+1}| -u^ 2_ 0-u^ 2_ 1+u^ 2_ 2+...+u^ 2_ n=-1\}\) be the anti de Sitter space with its induced Lorentz metric. Then the paper shows an isometric imbedding \(U_ n\hookrightarrow H^ n_ 1\) which is equivariant with respect to an isomorphism of the largest connected group of isometries of \(U_ n\) into the largest connected group SO(2,n-1) of isometries of \(H^ n_ 1\).
scientific article; zbMATH DE number 3861973