Megan M. Kerr - Academia.edu (original) (raw)

Papers by Megan M. Kerr

The Michigan Mathematical Journal, Apr 1, 1998

Transactions of the American Mathematical Society, 1996

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, ... more We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M) > 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we nd all G-homogeneous (non-symmetric) Einstein metrics on M.

Ann Glob Anal Geom, 1997

In this paper we examine homogeneous Einstein-Weyl structures and classify them on compact irredu... more In this paper we examine homogeneous Einstein-Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein-Weyl equation is very restrictive: Einstein-Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

... I thank David Harbater, Steve Shatz, and Frank Warner for encouragement throughout my graduat... more ... I thank David Harbater, Steve Shatz, and Frank Warner for encouragement throughout my graduate studies. I also thank Chris Croke and Herman Gluck for generously sharing their expertise. I am especially indebted to my advisor, Wolfgang Ziller, for so many hours of his time, ...

Annals of Global Analysis and Geometry, Mar 22, 2008

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, ... more We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on TpM decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H < K < G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by O. V. Manturov [Ma1, Ma2, Ma3] and J. Wolf [Wo1, Wo2].

Transactions of the American Mathematical Society

This paper is organized as follows: In section 1 we recall some basic facts about compact homogen... more This paper is organized as follows: In section 1 we recall some basic facts about compact homogeneous spaces. Then we give the classification of all simply connected compact homogeneous spaces M up to dimension n = 12, acted on by a compact, connected, simply connected and simple Lie group G. In the last section we provide a proof of our main theorem

Annals of Global Analysis and Geometry, 2014

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In [T... more We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In [T2], H. Tamaru constructs new attached solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to associated solvmanifolds, described in [GK], obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geometrically distinct from noncompact symmetric spaces and their submanifolds.

The Michigan Mathematical Journal, 1998

Proceedings of the American Mathematical Society, 2005

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature... more We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH 3 . This deformation includes a continuous family of new homogeneous Einstein spaces with negative sectional curvature.

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, ... more We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M) > 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we nd all G-homogeneous (non-symmetric) Einstein metrics on M.

We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we... more We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of tendimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetric spaces of rank greater than one, preserving the (constant) Ricci curvature.

Annals of Global Analysis and Geometry - ANN GLOB ANAL GEOM, 1997

In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irredu... more In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein–Weyl equation is very restrictive: Einstein–Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

Geometriae Dedicata, 2012

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate ... more We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.

Differential Geometry and its Applications, 2014

We classify the triples H ⊂ K ⊂ G of nested compact Lie groups which satisfy the "positive triple... more We classify the triples H ⊂ K ⊂ G of nested compact Lie groups which satisfy the "positive triple" condition that was shown in [17] to ensure that G/H admits a metric with quasi-positive curvature. A few new examples of spaces that admit quasi-positively curved metrics emerge from this classification; namely, a CP 2 -bundle over S 6 , a B 7 -bundle over HP 2 , a CP 2n−1 -bundle over HP n for each n 2, and a family of finite quotients of T 1 S 6 .

Annals of Global Analysis and Geometry, 2013

Annals of Global Analysis and Geometry, 2008

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, ... more We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on TpM decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H < K < G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by O. V. Manturov [Ma1, Ma2, Ma3] and J. Wolf [Wo1, Wo2].

Rocky Mountain Journal of Mathematics, 2010

We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We d... more We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We describe the connection and curvatures, and we investigate necessary and sufficient conditions for subgroups to be totally geodesic submanifolds. We also classify the one-parameter subgroups which are geodesics.

The Michigan Mathematical Journal, Apr 1, 1998

Transactions of the American Mathematical Society, 1996

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, ... more We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M) > 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we nd all G-homogeneous (non-symmetric) Einstein metrics on M.

Ann Glob Anal Geom, 1997

In this paper we examine homogeneous Einstein-Weyl structures and classify them on compact irredu... more In this paper we examine homogeneous Einstein-Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein-Weyl equation is very restrictive: Einstein-Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

... I thank David Harbater, Steve Shatz, and Frank Warner for encouragement throughout my graduat... more ... I thank David Harbater, Steve Shatz, and Frank Warner for encouragement throughout my graduate studies. I also thank Chris Croke and Herman Gluck for generously sharing their expertise. I am especially indebted to my advisor, Wolfgang Ziller, for so many hours of his time, ...

Annals of Global Analysis and Geometry, Mar 22, 2008

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, ... more We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on TpM decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H < K < G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by O. V. Manturov [Ma1, Ma2, Ma3] and J. Wolf [Wo1, Wo2].

Transactions of the American Mathematical Society

This paper is organized as follows: In section 1 we recall some basic facts about compact homogen... more This paper is organized as follows: In section 1 we recall some basic facts about compact homogeneous spaces. Then we give the classification of all simply connected compact homogeneous spaces M up to dimension n = 12, acted on by a compact, connected, simply connected and simple Lie group G. In the last section we provide a proof of our main theorem

Annals of Global Analysis and Geometry, 2014

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In [T... more We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In [T2], H. Tamaru constructs new attached solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to associated solvmanifolds, described in [GK], obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geometrically distinct from noncompact symmetric spaces and their submanifolds.

The Michigan Mathematical Journal, 1998

Proceedings of the American Mathematical Society, 2005

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature... more We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH 3 . This deformation includes a continuous family of new homogeneous Einstein spaces with negative sectional curvature.

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, ... more We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank(M) > 1, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup G of Isom(M) acting transitively on M we nd all G-homogeneous (non-symmetric) Einstein metrics on M.

We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we... more We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of tendimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetric spaces of rank greater than one, preserving the (constant) Ricci curvature.

Annals of Global Analysis and Geometry - ANN GLOB ANAL GEOM, 1997

In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irredu... more In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein–Weyl equation is very restrictive: Einstein–Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

Geometriae Dedicata, 2012

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate ... more We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.

Differential Geometry and its Applications, 2014

We classify the triples H ⊂ K ⊂ G of nested compact Lie groups which satisfy the "positive triple... more We classify the triples H ⊂ K ⊂ G of nested compact Lie groups which satisfy the "positive triple" condition that was shown in [17] to ensure that G/H admits a metric with quasi-positive curvature. A few new examples of spaces that admit quasi-positively curved metrics emerge from this classification; namely, a CP 2 -bundle over S 6 , a B 7 -bundle over HP 2 , a CP 2n−1 -bundle over HP n for each n 2, and a family of finite quotients of T 1 S 6 .

Annals of Global Analysis and Geometry, 2013

Annals of Global Analysis and Geometry, 2008

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, ... more We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on TpM decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H < K < G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by O. V. Manturov [Ma1, Ma2, Ma3] and J. Wolf [Wo1, Wo2].

Rocky Mountain Journal of Mathematics, 2010

We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We d... more We study the geometry of a filiform nilpotent Lie group endowed with a leftinvariant metric. We describe the connection and curvatures, and we investigate necessary and sufficient conditions for subgroups to be totally geodesic submanifolds. We also classify the one-parameter subgroups which are geodesics.