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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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
warped products; bilipschitz embeddings; negative curvature
Full Text:
##### 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
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