Pedro Ontaneda Portal is a Peruvian-Americanmathematician specializing intopology anddifferential geometry. He is a distinguished professor atBinghamton University, a unit of theState University of New York.^[1]

Ontaneda received his Ph.D. in 1994 from Stony Brook University (another unit of SUNY), advised byLowell Jones.^[2] Subsequently he taught at theFederal University of Pernambuco in Brazil.He moved to Binghamton University in 2005.

Ontaneda's work deals with the geometry and topology ofaspherical spaces, with particular attention to the relationship between exotic structures and negative or non-positive curvature on manifolds.

Classical examples ofRiemannian manifolds ofnegative curvature are given by realhyperbolic manifolds, or more generally bylocally symmetric spaces of rank 1.One of Ontaneda's most celebrated contributions is the construction of manifolds that admit negatively curved Riemannian metrics but do not admit locally symmetric ones.More precisely, he showed that for any${\displaystyle n\ge 4}$ and for any${\displaystyle \epsilon >0}$ there exists a closed Riemannian${\displaystyle n}$-manifold${\displaystyle N}$ satisfying the following two properties:^[3]

1. All thesectional curvatures of${\displaystyle N}$ are in${\displaystyle [-1-\epsilon ,-1]}$.
2. ${\displaystyle N}$ is not homeomorphic to a locally symmetric space.

In particular, the fundamental group of${\displaystyle N}$ isGromov hyperbolic but not isomorphic to a uniform lattice in a Lie group of rank 1.

These manifolds are obtained via theRiemannian hyperbolization procedure developed by Ontaneda in a series of papers, which is a smooth version of the strict hyperbolization procedure introduced byRuth Charney andMichael W. Davis.^[4] The obstruction to being locally symmetric comes from the fact that Ontaneda's manifolds have nontrivial rationalPontryagin classes.The restriction to dimension${\displaystyle n\ge 4}$ is necessary. Indeed, if a surface admits a negatively curved metric, then it admits one that is locally isometric to the realhyperbolic plane, as a consequence of theuniformization theorem. A similar statement holds for${\displaystyle 3}$-manifolds thanks to thehyperbolization theorem.

Ontaneda also made a "remarkable"^[5] contribution to the classification ofdynamical systems by constructing partially hyperbolic diffeomorphisms (a generalization ofAnosov diffeomorphisms) on some simply connected manifolds of high dimension; see his 2015 paper.

• F. T. Farrell, L. E. Jones, and P. Ontaneda (2007), "Negative curvature and exotic topology." InSurveys in Differential Geometry, Vol. XI, pp. 329–347, International Press, Somerville, MA.
• F. Thomas Farrell and Pedro Ontaneda (2010), "On the topology of the space of negatively curved metrics."Journal of Differential Geometry 86, no. 2, pp. 273–301.
• Andrey Gogolev, Pedro Ontaneda, and Federico Rodriguez Hertz (2015), "New partially hyperbolic dynamical systems I."Acta Mathematica 215, no. 2, pp. 363–393.
• Pedro Ontaneda (2020), "Riemannian hyperbolization."Publ. Math. Inst. Hautes Études Sci. 131, pp. 1–72.

• Pedro Ontaneda'sAuthor Profile on MathSciNet

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• American people of Peruvian descent
• American mathematicians
• Peruvian mathematicians
• Binghamton University faculty
• Topologists

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