[Paper Review] On M. Mueter's Ph.D. Thesis on Cheeger deformations

Wolfgang Ziller|ArXiv.org|Sep 1, 2009

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper summarizes Michael Müter's unpublished 1987 Ph.D. thesis on Cheeger deformations—a Riemannian metric deformation technique that shrinks metrics along group orbits while preserving non-negative curvature. Using a Riemannian submersion construction and curvature formulas involving the orbit tensor $ P_t $ and metric tensor $ C_t $, the paper shows that curvature behavior is best analyzed on moving planes via $ C_t^{-1} $, and proves that for $ ext{SU}(2) $ or $ \text{SO}(3) $ actions, any 2-plane with nontrivial orbit projection becomes positively curved for large $ t $, offering a simple proof of key results in positive curvature geometry.

ABSTRACT

These are notes of a talk I gave in a seminar at the University of Pennsylvania summarizing results in the Ph.D. thesis of Michael Mueter obtained under the direction of Wolfgang Meyer at the University of Muenster. His thesis on "Kruemmungserhoehende Deformationen mittels Gruppenaktionen" examines in detail curvature properties of so called Cheeger deformations. Such deformations have been a crucial ingredient in non-negative and positive curvature constructions. His thesis contains a wealth of interesting properties of such deformations, but since it was never published, is only known to a few experts. I have no intention to publish these notes, but post them as a service to the public. A scanned version of his thesis (written in German) is available on my homepage.

Motivation & Objective

• To make accessible the detailed curvature analysis of Cheeger deformations from Michael Müter's unpublished 1987 Ph.D. thesis, which remains a foundational but obscure resource.
• To clarify the behavior of sectional curvature under Cheeger deformations, particularly the non-intuitive fact that curvature of fixed planes can become negative, while curvature of moving planes $ \text{span}\{C_t^{-1}X, C_t^{-1}Y\} $ remains non-negative when the original metric has non-negative curvature.
• To provide a self-contained summary of Müter's results, including curvature formulas and geometric insights, for researchers working in Riemannian geometry and positive curvature metrics.
• To highlight the utility of Cheeger deformations in constructing and analyzing non-negatively and positively curved metrics on homogeneous spaces, biquotients, and cohomogeneity one manifolds.

Proposed method

• Constructs a Riemannian submersion $ \sigma: M \times G \to M $ via the action $ (p,g) \mapsto g^{-1}p $, inducing a one-parameter family of metrics $ g_t $ on $ M $.
• Defines the orbit tensor $ P_t $ via $ Q(P_t(X),Y) = g_t(X^*,Y^*)_p $ for $ X,Y \in \mathfrak{m}_p $, and derives $ P_t = P(I + tP)^{-1} $.
• Introduces the metric tensor $ C_t $ satisfying $ g(C_t(X),Y) = g_t(X,Y) $, with $ C_t(V) = (I + tP)^{-1}(V^\mathcal{V}) + V^\mathcal{H} $.
• Analyzes curvature by studying unnormalized sectional curvature $ \kappa_c(t) $, decomposed into intrinsic curvature, a term involving $ \|[P(V_\mathfrak{m}), P(W_\mathfrak{m})]\|_Q^2 $, and an O’Neill-type correction term $ z(V,W,t) $.
• Uses the transformation $ C_t^{-1} $ to track curvature of moving planes, showing that if the original metric has non-negative curvature, so do the images of such planes under $ C_t^{-1} $.
• Applies the formalism to prove that for $ \text{SU}(2) $ or $ \text{SO}(3) $ actions, all 2-planes with nontrivial projection onto orbits become positively curved as $ t \to \infty $.

Experimental results

Research questions

• RQ1 How does the sectional curvature of a 2-plane evolve under Cheeger deformation, especially when the original curvature is non-negative?
• RQ2 Why does the curvature of a fixed plane $ \text{span}\{X,Y\} $ sometimes become negative for $ t > 0 $, even if the original metric has non-negative curvature?
• RQ3 What is the geometric reason that curvature is better analyzed on the moving planes $ \text{span}\{C_t^{-1}X, C_t^{-1}Y\} $ rather than on fixed planes?
• RQ4 Under what conditions does a Cheeger deformation preserve or enhance non-negative curvature, particularly in the limit $ t \to \infty $?
• RQ5 Can Cheeger deformations be used to construct or prove the existence of positively curved metrics on homogeneous or biquotient spaces?

Key findings

• For $ \text{SU}(2) $ or $ \text{SO}(3) $ actions, any 2-plane with nontrivial projection onto the orbit becomes positively curved for sufficiently large $ t $, even if it had negative curvature initially.
• The unnormalized sectional curvature $ \kappa_c(t) $ is given by $ g(R(V,W),V,W) + \frac{t^3}{4}\|[P(V_\mathfrak{m}), P(W_\mathfrak{m})]\|_Q^2 + z(V,W,t) $, where $ z $ is an O’Neill-type correction term.
• The norm of the wedge product of the lifted vectors satisfies $ \|C_t^{-1}(V) \wedge C_t^{-1}(W)\|^2_{g_t} = t^2\|[P(V_\mathfrak{m}), P(W_\mathfrak{m})]\|_Q^2 + t(\|P(V_\mathfrak{m})\|^2 + \|P(W_\mathfrak{m})\|^2) + 1 $.
• Cheeger deformations with $ t < 0 $ can preserve non-negative curvature when scaling up biinvariant metrics along abelian subalgebras, provided $ t > -1/4 $, a fact derived from Müter’s formula but overlooked in earlier work.
• The space of homogeneous metrics with non-negative curvature on $ G/H $ is connected via Cheeger deformations, as the rescaled metric $ t g_t $ converges to the biinvariant metric as $ t \to \infty $.
• The inverse linear path $ t \mapsto t g_t $ provides a Taylor series expansion at the biinvariant metric, showing that only certain initial directions preserve non-negative curvature, and this path is used to study curvature stability.