A note on an isometric imbedding of upper half-space into the anti de Sitter space

Matsuda, Hiroo

doi:10.14492/hokmj/1381757712

Cited by 3 publications

(6 citation statements)

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“…The computation of the volume V n+1 (P ) is performed in a different model of DH n+1 1 built on the Minkowski space, which is based on a well known model of H n+1 1 (see Matsuda [4]). Some of the new properties of the model are explored in Section 3.…”

Section: Strategy Overviewmentioning

confidence: 99%

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A volume correspondence between anti-de Sitter space and its boundary

Zhang¹

2023

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Let H n+1 1 be the (n + 1)-dimensional anti-de Sitter space, in this paper we propose to extend the space by gluing it to another copy of H n+1 that is invariant under conformal symmetry, establishing a volume correspondence between DH n+1 1 and ∂H n+1 1 . 1 respectively) are consistent, in the sense that when DH n is treated as embedded in DH n+1 1 . Theorem 1.4. Let P ∈ H 0 in DH n+1 1, then V n+1 (P ) is real for n odd, and V n+1 (P ) is imaginary for n even and is completely determined by P ∩ ∂H n+1 1 .

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Section: Strategy Overviewmentioning

confidence: 99%

“…1 For n ≥ 1, Matsuda [4] used an upper half-space U n+1 to partially represent H n+1 . A similar representation of the de Sitter space was given by Nomizu [7].…”

Section: A New Model For Dh N+1mentioning

confidence: 99%

A volume correspondence between anti-de Sitter space and its boundary

Zhang¹

2023

Preprint

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“…n ) 1/2 /x 1 and an isometric embedding of U n into the anti-de Sitter space (see Matsuda [7] for the corresponding formula f for the isometric embedding), and eventually get the same extension of R n−1,1 that is also conformally equivalent to the extension of the anti-de Sitter space except at a codimension 1 region. This also suggests that the extension of anti-de Sitter is conformally equivalent to the double de Sitter space except at a codimension 1 region (recall Remark 8.1 that the extension of anti-de Sitter space is also topologically S n−1 × S 1 , the same as the double de Sitter space DS n 1 ).…”

Section: An Extension Of R Pqmentioning

confidence: 99%

“…In this paper the term "double cover" is used loosely to only refer to the open part of a space, usually with the boundary points ignored, and also the metrics on the two covers need not agree 7. The sign of ℓ(x) has more to do with the linear space (R n,1 or R n,1 − ) the sheet is embedded in than with the sheet itself, but for now this definition of ℓ(x) is convenient and enough for our purposes.…”

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confidence: 99%

On the total volume of the double hyperbolic space

Zhang

2018

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Let the double hyperbolic space DH n , proposed in this paper as an extension of the hyperbolic space H n and homeomorphic to the standard unit n-sphere S n , contain both the two sheets of the two-sheeted hyperboloid (but with the geodesics in the lower sheet H n − negatively valued, and so is the associated volume element of H n − if n is odd), which are connected at the boundary at infinity by identifying ∂H n = ∂H n − projectively. We propose to extend the volume of the polytopes in H n to polytopes in DH n , where the volume can possibly be complex valued. With some mild assumption, we prove a Schläfli differential formula for DH n , and show that the total volume of DH n is equal to i n V n (S n ) for both even and odd dimensions. When n = 2m + 1, on top of the conformal structure of ∂H 2m+1 it also induces an intrinsic volume on regions in ∂H 2m+1 that we call polytopes, and we explore a Schläfli differential formula for ∂H 2m+1 as well. As a dual of DH n , an extension of the de Sitter space is proposed, which is compact without boundary and homeomorphic to S n−1 × S 1 ; and it also induces an extension of R n−1,1 , reminiscent of the conformal compactification of spacetime.

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“…(vi) M = U~ = {(ui, ■ ■ ■ ,un); un > 0} with metric ds2_ = (-du\ + du?, + ■ ■ ■ + du2n)/(cun)2 (c ¿ 0), and G = I°(U~) (see Matsuda [3]). …”

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confidence: 99%