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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.

53C20 Global Riemannian geometry, including pinching
Full Text: DOI
[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
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