×

zbMATH — the first resource for mathematics

Bilipschitz embeddings of negative sectional curvature in products of warped product manifolds. (English) Zbl 1012.53033
The authors generalize results of N. Brady and B. Farb [Trans. Am. Math. Soc. 350, 3393-3405 (1998; Zbl 0920.53023)] to show the existence of bilipschitz embedded manifolds \(i:Y\rightarrow X\) of negative sectional curvature in Riemannian products of certain types of warped manifolds. The base \(B\) is assumed either to be \(1\) dimensional or a Riemannian manifold of arbitrary dimension with negative sectional curvature. The warping functions \(f_i:B\rightarrow R^+\) are assumed to be strictly convex functions without minimum so that \(\operatorname {grad} f_i(f_j)>0\) for all \(i,j\). The fibers \(F_i\) are assumed to be either one dimensional or to have nonpositive sectional curvature. Let \(M_i:=B\times F_i\) be given the warped product metrics \(ds^2_{M_i}:=ds^2_B+f_i^2ds^2_{F_i}\). Let \(X=M_1\times \dots \times M_k\) be given the product metric. Let \(Y:=B\times F_1\times\dots F_k\). Define an embedding \(i:Y\rightarrow X\) by setting \(i(b,\xi_1,\dots ,\xi_k):=(b,\xi_1)\times\dots \times(b,\xi_k)\). The author shows that \(ds^2_Y:=i^*ds^2_X\) is a metric of negative sectional curvature on \(Y\). The author also shows that if all the warping functions \(f_i\) are the same, then \(i\) is a bilipschitz embedding.

MSC:
53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1 – 49. · Zbl 0191.52002
[2] Noel Brady and Benson Farb, Filling-invariants at infinity for manifolds of nonpositive curvature, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3393 – 3405. · Zbl 0920.53023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.