Riemannian manifolds with nontrivial Limbeek local symmetry Wouter - PowerPoint PPT Presentation

Riemannian manifolds with nontrivial local symmetry Wouter van Riemannian manifolds with nontrivial Limbeek local symmetry Wouter van Limbeek University of Chicago limbeek @ math.uchicago.edu 21 October 2012

The problem Riemannian manifolds with nontrivial local symmetry Wouter van Limbeek Let M be a closed Riemannian manifold. Isom( ˜ M ) contains the deck group π 1 ( M ). Generically: [ Isom ( ˜ M ) : π 1 M ] < ∞ . Problem Classify M such that [ Isom ( ˜ M ) : π 1 M ] = ∞ .

Example Riemannian manifolds with nontrivial local symmetry M closed hyperbolic manifold. Wouter van Limbeek Theorem (Bochner, Yano) Isom(M) is finite. But Isom ( ˜ M ) = Isom ( H n ) = O + ( n , 1) . Note: ˜ M is homogeneous.

Farb-Weinberger theorem Riemannian manifolds with nontrivial Theorem (Farb, Weinberger (2008)) local symmetry Let M be a closed aspherical manifold. Then either Wouter van Limbeek 1 [ Isom ( ˜ M ) , π 1 M ] < ∞ or 2 M is on a list. Further, every item on the list occurs.

Farb-Weinberger theorem Riemannian manifolds with nontrivial Theorem (Farb, Weinberger (2008)) local symmetry Let M be a closed aspherical manifold. Then either Wouter van Limbeek 1 [ Isom ( ˜ M ) , π 1 M ] < ∞ or 2 M is on a list. Further, every item on the list occurs. Applications 1 Differential geometry 2 Complex geometry 3 etc.

The list Riemannian A finite cover of M is a ‘Riemannian orbibundle’ manifolds with nontrivial F → M ′ → B . local symmetry The fibers F are locally homogeneous. Wouter van Limbeek [ Isom (˜ B ) : π 1 B ] < ∞ .

The general (nonaspherical) case Riemannian manifolds with nontrivial Problems in general case local symmetry Proof of Farb and Weinberger fails: Wouter van M aspherical ⇒ Limbeek (geometry of ˜ M ) ↔ (geometry of π 1 ( M ) ) .

The general (nonaspherical) case Riemannian manifolds with nontrivial Problems in general case local symmetry Proof of Farb and Weinberger fails: Wouter van M aspherical ⇒ Limbeek (geometry of ˜ M ) ↔ (geometry of π 1 ( M ) ) . Crucial in F-W: Isom( ˜ M ) 0 -orbits are of the same type. Not true in general case.

The general (nonaspherical) case Riemannian manifolds with nontrivial Problems in general case local symmetry Proof of Farb and Weinberger fails: Wouter van M aspherical ⇒ Limbeek (geometry of ˜ M ) ↔ (geometry of π 1 ( M ) ) . Crucial in F-W: Isom( ˜ M ) 0 -orbits are of the same type. Not true in general case. More options for Isom( ˜ M ): E.g. compact factors.

The general (nonaspherical) case Riemannian manifolds with nontrivial Problems in general case local symmetry Proof of Farb and Weinberger fails: Wouter van M aspherical ⇒ Limbeek (geometry of ˜ M ) ↔ (geometry of π 1 ( M ) ) . Crucial in F-W: Isom( ˜ M ) 0 -orbits are of the same type. Not true in general case. More options for Isom( ˜ M ): E.g. compact factors. So the ‘list’ is more complicated.

More complicated example Riemannian manifolds with nontrivial Example local symmetry Wouter van Limbeek     1 ∗ ∗ 1 0 ∗  , Z ( H ) = H := 0 1 0 1 0 ∗    0 0 1 0 0 1

More complicated example Riemannian manifolds with nontrivial Example local symmetry Wouter van Limbeek     1 ∗ ∗ 1 0 ∗  , Z ( H ) = H := 0 1 0 1 0 ∗    0 0 1 0 0 1 Set N := H / Z .

More complicated example Riemannian manifolds with nontrivial Example local symmetry Wouter van Limbeek     1 ∗ ∗ 1 0 ∗  , Z ( H ) = H := 0 1 0 1 0 ∗    0 0 1 0 0 1 Set N := H / Z . = S 1 act on S 2 by rotations. Let Z ( N ) ∼

More complicated example Riemannian manifolds with nontrivial Example local symmetry Wouter van Limbeek     1 ∗ ∗ 1 0 ∗  , Z ( H ) = H := 0 1 0 1 0 ∗    0 0 1 0 0 1 Set N := H / Z . = S 1 act on S 2 by rotations. Let Z ( N ) ∼ Let X := ( S 2 × N ) / Z ( N ).

More complicated example Riemannian manifolds with nontrivial Example local symmetry Wouter van Limbeek     1 ∗ ∗ 1 0 ∗  , Z ( H ) = H := 0 1 0 1 0 ∗    0 0 1 0 0 1 Set N := H / Z . = S 1 act on S 2 by rotations. Let Z ( N ) ∼ Let X := ( S 2 × N ) / Z ( N ). Orbits in X are of two types: N (generic) 1 N / S 1 = R 2 (north/south poles) 2

General fact about Lie groups Riemannian manifolds with nontrivial local symmetry Theorem (Levi decomposition) Wouter van Let G be a connected Lie group. Then Limbeek There exists a solvable subgroup G sol and there exists a semisimple subgroup G ss such that G = G sol G ss . Remark This decomposition is essentially unique.

Nonaspherical case Riemannian manifolds with nontrivial local symmetry Theorem (VL) Wouter van Limbeek Let M be a closed Riemannian manifold, G := Isom ( ˜ M ) . Then either

Nonaspherical case Riemannian manifolds with nontrivial local symmetry Theorem (VL) Wouter van Limbeek Let M be a closed Riemannian manifold, G := Isom ( ˜ M ) . Then either 1 G 0 is compact or

Nonaspherical case Riemannian manifolds with nontrivial local symmetry Theorem (VL) Wouter van Limbeek Let M be a closed Riemannian manifold, G := Isom ( ˜ M ) . Then either 1 G 0 is compact or 2 G 0 ss is compact and G has infinitely many components or

Nonaspherical case Riemannian manifolds with nontrivial local symmetry Theorem (VL) Wouter van Limbeek Let M be a closed Riemannian manifold, G := Isom ( ˜ M ) . Then either 1 G 0 is compact or 2 G 0 ss is compact and G has infinitely many components or 3 M is on a ‘list’.

Nonaspherical case: The list Riemannian manifolds with nontrivial local The list symmetry Wouter van Limbeek

Nonaspherical case: The list Riemannian manifolds with nontrivial local The list symmetry 1 G 0 Wouter van ss noncompact ⇒ Limbeek M ‘virtually’ fibers over locally symmetric space.

Nonaspherical case: The list Riemannian manifolds with nontrivial local The list symmetry 1 G 0 Wouter van ss noncompact ⇒ Limbeek M ‘virtually’ fibers over locally symmetric space. 2 G 0 ss is compact, #( components of G ) < ∞ ⇒ M is ‘virtually’ an ‘iterated bundle’ over tori.

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial local symmetry Wouter van Limbeek

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group local symmetry Wouter van Limbeek

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group � Map f 1 : M → N local symmetry Wouter van Limbeek

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group � Map f 1 : M → N local symmetry Wouter van Γ starts ‘tower of lattices’ (Γ q ) q Limbeek

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group � Map f 1 : M → N local symmetry Wouter van Γ starts ‘tower of lattices’ (Γ q ) q � Map f q : M q → N q Limbeek

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group � Map f 1 : M → N local symmetry Wouter van Γ starts ‘tower of lattices’ (Γ q ) q � Map f q : M q → N q Limbeek Limit ˜ f : ˜ M → ˜ Arzel` a-Ascoli � N

G 0 is nilpotent: Outline Riemannian Proof. manifolds with nontrivial Γ ⊆ G 0 lattice in nilpotent group � Map f 1 : M → N local symmetry Wouter van Γ starts ‘tower of lattices’ (Γ q ) q � Map f q : M q → N q Limbeek Limit ˜ f : ˜ M → ˜ Arzel` a-Ascoli � N Smooth ˜ f while keeping it equivariant.

G 0 ss noncompact: Outline Riemannian manifolds with nontrivial local symmetry Wouter van Find a lattice Λ in a semisimple Lie group Limbeek and a map Γ → Λ. � homotopy class of maps M → N ( N locally symmetric space for Λ). Theorem (Eells, Sampson, Hartman, Schoen-Yau) ∃ ! harmonic f : M → N in this class.

G 0 ss noncompact: Outline Riemannian manifolds with nontrivial local Lift to ˜ f : ˜ M → ˜ N . symmetry Wouter van Limbeek Theorem (Frankel, 1994) One can average ˜ f . � the fiber bundle M → N . Remarks Frankel’s method relies heavily on symmetric space theory. This does not work if G 0 ss is compact.

Open question Riemannian manifolds with nontrivial Question local symmetry Let M be a closed Riemannian manifold. Is it true that either Wouter van Limbeek M ) 0 is compact or 1 Isom( ˜ 2 M is virtually an iterated orbibundle, at each step with locally homogeneous fibers or base? More specifically: Problem Describe closed Riemannian manifolds M such that G has infinitely many components and G 0 ss is compact.