Non-compact Einstein manifolds with symmetry

Abstract

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with compact, smooth orbit space, we show that the nilradical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts polarly and that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.

Topics & Concepts