There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Tables of integrals

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

The observation of the inspiral and merger of compact binaries by the LIGO/Virgo collaboration ushered in a new era in the study of strong-field gravity. We review current and future tests of strong gravity and of the Kerr paradigm with gravitational-wave interferometers, both within a theory-agnostic framework (the parametrized post-Einsteinian formalism) and in the context of specific modified theories of gravity (scalar–tensor, Einstein–dilaton–Gauss–Bonnet, dynamical Chern–Simons, Lorentz-violating, and extra dimensional theories). In this contribution we focus on (i) the information carried by the inspiral radiation, and (ii) recent progress in numerical simulations of compact binary mergers in modified gravity.

Extreme gravity tests with gravitational waves from compact binary coalescences: (I) inspiral–merger

Manifolds all of whose geodesics are closed

ELLIPTIC OPERATORS Introduction The Real and Complex Laplace Operators Spinors Spectral Resolutions Manifolds with Boundary Spectral Invariants The Eta Invariant Computing the Eta Invariant DIFFERENTIAL GEOMETRY Introduction Riemannian Submersions Characteristic Classes The Geometry of Sphere and Principal Bundles The Geometry of Circle Bundles The Hopf Fibration The Scalar Curvature Levi-Civita and Spin Connections POSITIVE CURVATURE Introduction Manifolds with Positive Ricci Curvature Bordism and Connective K Theory Calculations Involving the Eta Invariant The Eta Invariant and Connective K Theory Computing Connective K Theory Groups SPECTRAL GEOMETRY OF RIEMANNIAN SUBMERSIONS Introduction Intertwining the Coderivitives The Real Laplacian The Complex Laplacian The Spin Laplacian Riemannian Submersions with Boundary Heat Trace and Heat Content Unresolved Questions REFERENCES Introduction Main Bibliography Bibliography of Harmonic Morphisms Parabolic PDE Bibliography NOTATION INDEX

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

In this paper, we study the geometry of anti-invariant Riemanniansubmersions from a Kahler manifold onto a Riemannian manifold. Wefirst determine the base space when the total space of ananti-invariant Riemannian submersion is Einstein and then weinvestigate new conditions for anti-invariant Riemannian submersionsto be Clairaut submersions. We also focus on the geometry ofClairaut anti-invariant submersions.

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Einstein conditions for the base space of anti-invariant riemannian submersions and clairaut submersions

We recall a curvature identity for 4-dimensional compact Riemannian manifolds as derived from the generalized Gauss–Bonnet formula. We extend this curvature identity to non-compact 4-dimensional Riemannian manifolds. We also give some applications of this curvature identity.

A Curvature Identity on a 4-Dimensional Riemannian Manifold

Let \(Y\) be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let \(\pi:P\rightarrow Y\) be a principal bundle over \(Y\) with compact connected structure group \(G\). If the fundamental group of \(P\) is finite, we show that \(P\) admits a \(G\) invariant metric with positive Ricci curvature so that \(\pi\) is a Riemannian submersion.

Invariant metrics of positive Ricci curvature on principal bundles

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss–Bonnet theorem using heat equation methods, to give a new proof of a result of Kuzʼmina and Labbi concerning the Euler–Lagrange equations of the Gauss–Bonnet integral, and to give a new derivation of the Euh–Park–Sekigawa identity.

JeongHyeong Park

Papers

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

Einstein conditions for the base space of anti-invariant riemannian submersions and clairaut submersions

A Curvature Identity on a 4-Dimensional Riemannian Manifold

Invariant metrics of positive Ricci curvature on principal bundles

Universal curvature identities