algebraic and differential geometry. computational...

Symmetric-like manifoldsApplications in Inverse Spectral Geometry

Modifying the classical setting: The Steklov spectrum

Algebraic and Differential Geometry.Computational Algebra.

Teresa Arias-Marco

Department of MathematicsUniversity of Extremadura, Badajoz, Spain

January 14th, 2016

First Joint Meeting Evora-Extremaduraon Mathematics

Teresa Arias-Marco GADAC

The research group GADACweb page http://www.unex.es/investigacion/grupos/gadacGroup Coordinator IGNACIO OJEDAResearch Lines of IGNACIO OJEDA

Semigroup algebras. Toric geometry.Classification of projective curves.

Group Members and Research lines

Amelia Alvarez – Schemes algebras and their representations.Teresa Arias-Marco–Homogeneous spaces with additionalgeometric structures.Adrian Gordillo–Natural tensors, differential invariants andRiemannian metrics.Jose Navarro–Natural tensors, differential invariants andRiemannian metrics.Juan A. Navarro GonzalezPedro Sancho de Salas – Schemes algebras and their rep.David Sevilla–Algorithms for algebraic varieties.

Amelia Alvarez – Schemes algebras and their representations.Teresa Arias-Marco–Homogeneous spaces withadditional geometric structures.Adrian Gordillo–Natural tensors, differential invariants andRiemannian metrics.Jose Navarro–Natural tensors, differential invariants andRiemannian metrics.Juan A. Navarro GonzalezPedro Sancho de Salas – Schemes algebras and their rep.David Sevilla–Algorithms for algebraic varieties.

Homogeneous spaces with additional geometric structures

T. Arias-Marco, D. Schueth.On inaudible curvature properties of closed Riemannianmanifolds.Ann. Glob. Anal. Geom., (2010), 37:339-349.

T. Arias-Marco, D. Schueth.Local symmetry of harmonic spaces as determined by thespectra of small geodesic spheres.Geom. Funct. Anal. (GAFA), (2012), 22:1-21.

1 Symmetric-like manifolds

2 Applications in Inverse Spectral GeometryThe settingOur Results and open problems

3 Modifying the classical setting: The Steklov spectrum

Einstein manifolds and Local homogeneity

Definition

(M, g) is an Einstein manifold ifric = C · g for a certain constant Cat each point of M.

Definition.(M, g) is an homogeneous

Riemannian manifold if I(M)acts transitively on M.

Definition

(M, g) is locally homogeneous if thepseudogroup of local isometries actstransitively on M.

Einstein manifolds and Local homogeneity

Definition

(M, g) is an Einstein manifold ifric = C · g for a certain constant Cat each point of M.

Definition.(M, g) is an homogeneous

Riemannian manifold if I(M)acts transitively on M.

Definition

(M, g) is locally homogeneous if thepseudogroup of local isometries actstransitively on M.

Spaces of type A

Definition (A. Gray, 1978)

(M, g) is of type A (TAm) ifric is cyclic-parallel; i.e.(∇X ric)(X ,X ) = 0 for all X ∈ TM.

Szabo 1993. TAm are analytic.

D’Atri, Nickerson 1969. TAmhave constant scalar curvature.

Kowalski 1983, Pedersen, Tod1999. TAm up to dimension 3are homogeneous and classified.

-, Kowalski 2008. Classifiedhomogeneous TAm of dim 4.

Pedersen, Tod 1999. Indimensions ≥ 5, there existnon-locally homogeneous TAm.

Tod 1999. In dim 4, TAmcondition doesn’t imply localhomogeneity.

Spaces of type A

Symmetric spaces

Definition. sp at p ∈ M is a local geodesic symmetry if for allX ∈ N(p) in TpM, sp(expp(X )) = expp(−X ).

Definition.

(M, g) is locally symmetric if for allp ∈ M, sp is an isometry.

≡ ∇R = 0.

Definition

(M, g) is a Riemannian symmetricspace if for all p ∈ M, sp is globallydefined and it is an isometry.

⇒ Riemannian symmetric spaces are

spaces with metric - preserving

geodesic symmetries.

Symmetric spaces

Definition. sp at p ∈ M is a local geodesic symmetry if for allX ∈ N(p) in TpM, sp(expp(X )) = expp(−X ).

Definition.

(M, g) is locally symmetric if for allp ∈ M, sp is an isometry.

≡ ∇R = 0.

Definition

(M, g) is a Riemannian symmetricspace if for all p ∈ M, sp is globallydefined and it is an isometry.

⇒ Riemannian symmetric spaces are

spaces with metric - preserving

geodesic symmetries.

Harmonic Spaces

Definition

(M, g) is an harmonic space iff

the volume density function ofthe geodesic exponential map isradial around each point.

at each m ∈ M there exists anormal neighborhood of M onwhich ∆u = 0 admits a realsolution depending only uponthe distance r to m and beinganalytic for r 6= 0.

Besse, 1978: Every Harmonicmanifold is Einstein.

Open problem: Are harmonicspaces always locallyhomogeneous?

Harmonic Spaces

Definition

(M, g) is an harmonic space iff

the volume density function ofthe geodesic exponential map isradial around each point.

at each m ∈ M there exists anormal neighborhood of M onwhich ∆u = 0 admits a realsolution depending only uponthe distance r to m and beinganalytic for r 6= 0.

Besse, 1978: Every Harmonicmanifold is Einstein.

Open problem: Are harmonicspaces always locallyhomogeneous?

Weakly symmetric spaces

Definition (Selberg, 56, Szabo, 93)

(M, g) is called weakly symmetric ifeach p ∈ M and each nontrivial γstarting in p there exists an isometryof M which fixes p and reverses γ.

Berndt,Vanhecke,96. Weaklysym.examples non-symmetric.

Classifications3, 4-dim by Berndt,Vanhecke,96.5-dim by Kowalski,Marinosci,97.

Weakly symm. spaces are:Homogeneous by Selberg,56.Type A by Kowalski, Prufer,89.

Remark.

If a Riemannian covering of(M, g) is weakly symmetric

⇓(M, g) is Type A and

locally weakly symmetric.

Remark.

Weakly locally symmetric spaces

Definition

(M, g) is weakly locally symmetric if∀p ∈ M, ∃ε > 0 such that ∀γ in M,γ(0) = p, ∃ an isometry of Bε(p)which fixes p and reverses γ|(−ε,ε).

(M, g) is locally symmetric

⇓(M, g) is Weakly locally sym.

Weakly symmetric examples byBerndt,Vanhecke,96. are alsonon locally symmetric.

Lemma.Let (M, g) be complete, simplyconnected and, weakly locallysymmetric V M is weakly sym.

Complete Weakly locallysymmetric spaces are spacesof type A.

Definition

Others symmetric-like manifolds

Definition (D’Atri,Nickerson,69,74)

(M, g) is called D’Atri space if thelocal geodesic symmetries areRiemannian volume-preserving.

On complete Riemannianmanifolds

Weak local symmetry⇓

D’Atri property,

the C property,

probabilistic commutativity,

the TC property,

the GC property,

⇓The type A property.

Definition (Berndt,Vanhecke,92)

(M, g) is called C-space if for each γthe eigenvalues of the associatedJacobi operator are constant along γ.

D’Atri property,

the C property,

the TC property,

the GC property,

Definition (Kowalski, Prufer, 82, 89)

(M, g) is probabilistic commutativeif all Euclidean Laplacian ∆(k),k ∈ N, commute.

D’Atri property,

the C property,

the TC property,

the GC property,

Definition (Berndt, Vanhecke, 1993)

(M, g) is a TC-space if ∀m ∈ M and∀p ∈ M suff. close to m, Tp(m) andTsm(p)(m) have the same eigenvalues

D’Atri property,

the C property,

the TC property,

the GC property,

AM Teresa Arias-Marco GADAC

Definition

(M, g) is a GC-space if ∀m ∈ M and∀p ∈ M suff. close to m, Tm(p) ands−1m∗ ◦Tm(sm(p)) ◦ sm∗ have the sameeigenvalues

D’Atri property,

the C property,

the TC property,

the GC property,

AM Teresa Arias-Marco GADAC

Open problems on symmetric-like manifolds

Eigenvalues problems for the Laplace-Beltrami operator

Let (M, g) be a compact, connected, n-dimensional smoothRiemannian manifold, possibly with boundary.

Let ∆ = −div grad be the Laplace operator associated with g ,acting on functions.

Eigenvalues Problems: Find all real numbers λ for which thereexists a nontrivial solution f ∈ C2(M) to ∆f + λf = 0 when

∂M = ∅. Namely, closed eigenvalue problem.

f = 0 on ∂M. Namely, Dirichlet eigenvalue problem.

∂ν f = 0 on ∂M where ν is the inward unit normal vector fieldon ∂M. Namely, Neumann eigenvalue problem.

Throughout our work, (M, g) will be a closed manifold.

The spectrum of ∆ and the geometry of M

Definition. The heat kernel K (t, x , y) is the fundamentalsolution of ( ∂∂t + ∆)f = 0. K (t, x , y) is analytic in t > 0 andC∞ in (x , y) ∈ M ×M.

The Sturm-Liouville decomposition:∃ a complete orthonormal basis {ϕ0, ϕ1, ....} of C∞(M), with ϕj

having eigenvalue λj of ∆ satisfying 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞.Each eigenvalue has finite multiplicity. Finally,

K (t, x , y) =∞∑j=0

e−λj tϕj(x)ϕj(y)

with convergence absolute, and uniform, for each t > 0. Inparticular, ∫

MK (t, x , x)dV (x) =

∞∑j=0

e−λj t .

Definition. The heat kernel K (t, x , y) is the fundamentalsolution of ( ∂∂t + ∆)f = 0. K (t, x , y) is analytic in t > 0 andC∞ in (x , y) ∈ M ×M.

The Sturm-Liouville decomposition:∃ a complete orthonormal basis {ϕ0, ϕ1, ....} of C∞(M), with ϕj

having eigenvalue λj of ∆ satisfying 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞.Each eigenvalue has finite multiplicity. Finally,

K (t, x , y) =∞∑j=0

e−λj tϕj(x)ϕj(y)

with convergence absolute, and uniform, for each t > 0. Inparticular, ∫

MK (t, x , x)dV (x) =

∞∑j=0

e−λj t .

Minakshisundaram-Pleijel Asymptotic Expansion, 1949:

K (t, x , x) v (4πt)−n2

∞∑k=0

uk(x)tk

where uk(x) are C∞(M) and polynomials in the components of thecurvature tensor R and its covariant derivatives. Moreover,

∞∑j=0

e−λj t v (4πt)−n2

∞∑k=0

aktk ,

where the coefficients ak are called classical heat invariants. Inparticular,

∫Mω = Vol(M, g), a1 =

∫Mτω,

(2|R|2 − 2|ric|2 + 5τ2)ω, ...

where τ denote the scalar curvature.Teresa Arias-Marco GADAC

Minakshisundaram-Pleijel Asymptotic Expansion, 1949:

K (t, x , x) v (4πt)−n2

∞∑k=0

uk(x)tk

where uk(x) are C∞(M) and polynomials in the components of thecurvature tensor R and its covariant derivatives. Moreover,

∞∑j=0

e−λj t v (4πt)−n2

∞∑k=0

aktk ,

where the coefficients ak are called classical heat invariants. Inparticular,

∫Mω = Vol(M, g), a1 =

∫Mτω,

(2|R|2 − 2|ric|2 + 5τ2)ω, ...

where τ denote the scalar curvature.Teresa Arias-Marco GADAC

V The spectrum of ∆ depends only on the Riemannianstructure of (M, g).

Direct and inverse problems on spectral theory.

Direct: To use geometrical properties of M to determineinformation about the eigenvalues and eigenfunctions of ∆.

Theorem (Lichnerowicz)

(M, g) closed and ric ≥ (n − 1)k > 0, for k ∈ N.Then, the first positive eigenvalue λ satisfies λ ≥ nk.

Inverse: one assumes knowledge about Spec(∆,M) and attemptsto determine information about the geometry of (M, g).

Inverse problems in spectral geometry

To which extent does the spectrum of ∆ on (M, g)determine its geometry?

Important fact:

If (M, g) is isometric to (M ′, g ′) V Spec(∆,M) = Spec(∆′,M ′).

Natural question:

Does Spec(∆,M) = Spec(∆′,M ′) imply M isometric to M ′?

Negative answer, 1964: Milnor’s example(a pair of 16-dim flat tori)

Famous question by M. Kac, 1966:

Can one hear the shape of a drum?

Partial positive result: Spec(∆,M) = Spec(∆′, a ball) implies M isa ball of the same radius so isometric to that ball.

Important fact:

Natural question:

Important fact:

Natural question:

Important fact:

Natural question:

Basic definitions

Definition

Two compact Riemannian manifolds (M, g) and (M ′, g ′) are saidto be isospectral if Spec(∆,M) = Spec(∆′,M ′).

Definition

A geometric property of a compact Riemannian manifold M can beheard if it can be determined from Spec(∆,M).

Definition

A geometric property is inaudible, i.e. not determined by thespectrum, if there exist pairs of isospectral manifolds which differwith respect to this property.

Basic definitions

Definition

Two compact Riemannian manifolds (M, g) and (M ′, g ′) are saidto be isospectral if Spec(∆,M) = Spec(∆′,M ′).

Definition

A geometric property of a compact Riemannian manifold M can beheard if it can be determined from Spec(∆,M).

Definition

A geometric property is inaudible, i.e. not determined by thespectrum, if there exist pairs of isospectral manifolds which differwith respect to this property.

Some Positive and Negative Results

Some positive results:

Heat invariants can be heard:

∫Mω = Vol(M, g), a1 =

∫Mτω,

(2|R|2 − 2|ric|2 + 5τ2)ω, ...

V Therefore, in dim 2, the total curvature and the Eulercharacteristic of the manifold can be heard.

Some negative results:

Gordon, Gornet, Schueth, Webb, Wilson, 1998:

The maximum of the scalar curvature τ is inaudible.

Schueth, 1999: ∫M |ric|

2 is inaudible.

∫Mω = Vol(M, g), a1 =

∫Mτω,

(2|R|2 − 2|ric|2 + 5τ2)ω, ...

2 is inaudible.

∫Mω = Vol(M, g), a1 =

∫Mτω,

(2|R|2 − 2|ric|2 + 5τ2)ω, ...

2 is inaudible.

Our Results

Main Theorem (Ann. Glob. Anal. Geom. 37(2010), 339–349.)

Each of the following properties is an inaudible property ofRiemannian manifolds:

Weak local symmetry,

D’Atri property,

the C property,

the TC property,

the GC property,

the type A property.

Our Results

D’Atri property,

the C property,

the TC property,

the GC property,

Our Results

D’Atri property,

the C property,

the TC property,

the GC property,

An interesting question

Problem by B.Y. Chen, L. Vanhecke in 1981:

To what extent do the properties of sufficiently small geodesicspheres determine the Riemannian geometry of the ambient space?

Tp(m):Shape op.(at m)of the geodesic sph. with center p, radius r

Characterization (Vanhecke, Willmore, 1983)

M is locally symmetric space iff ∀m ∈ M and ∀p ∈ Msufficiently close to m, Tp(m) = Tsm(p)(m).

Characterization (Ledger, Vanhecke, 1987)

M is locally symmetric space iff ∀m ∈ M and ∀p ∈ Msufficiently close to m, sm∗ ◦ Tm(p) = Tm(sm(p)) ◦ sm.

↗↘TP

↗↘GP

An interesting question

Problem by B.Y. Chen, L. Vanhecke in 1981:

To what extent do the properties of sufficiently small geodesicspheres determine the Riemannian geometry of the ambient space?

Tp(m):Shape op.(at m)of the geodesic sph. with center p, radius r

Characterization (Vanhecke, Willmore, 1983)

M is locally symmetric space iff ∀m ∈ M and ∀p ∈ Msufficiently close to m, Tp(m) = Tsm(p)(m).

Characterization (Ledger, Vanhecke, 1987)

M is locally symmetric space iff ∀m ∈ M and ∀p ∈ Msufficiently close to m, sm∗ ◦ Tm(p) = Tm(sm(p)) ◦ sm.

↗↘TP

↗↘GP

Open problems

Question 1

Are the properties of being locally symmetric, Einstein or harmonicspectrally determined on closed manifold?

Question 2

To what extent do the spectra of small geodesic spheres in a(possibly noncompact) Riemannian manifold M determine thegeometry of M ?

Question 3

Is the property of being locally symmetric determined by thespectrum of sufficiently small geodesic spheres?

Open problems

Question 1

Question 2

Question 3

Open problems

Question 1

Question 2

Question 3

Our question and result on geodesic spheres

Our Question

Do the spectra of geodesic spheres distinguish symmetric(∇R = 0) harmonic spaces from nonsymmetric (∇R 6= 0)harmonic spaces?

Main Theorem (Geom. Funct. Anal. (GAFA) 22(2012), 1–21.)

Let M1 and M2 be harmonic spaces, and let p1 ∈ M1, p2 ∈ M2.If there exists ε > 0 such that for each r ∈ (0, ε) the geodesicspheres Sr (p1) and Sr (p2) are isospectral, then

|∇R|2p1 = |∇R|2p2

Consequently,

M1 is locally symmetric if and only if M2 is locally symmetric.

Our Question

|∇R|2p1 = |∇R|2p2

Consequently,

Our Question

|∇R|2p1 = |∇R|2p2

Consequently,

Our question and result on geodesic balls

Our Question

Do the spectra of geodesic balls distinguish symmetric (∇R = 0)harmonic spaces from nonsymmetric (∇R 6= 0) harmonic spaces?

Let M1 and M2 be harmonic spaces, and let p1 ∈ M1, p2 ∈ M2.If there exists ε > 0 such that for each r ∈ (0, ε) the geodesic ballsBr (p1) and Br (p2) are Dirichlet or Neumann isospectral, then

|∇R|2p1 = |∇R|2p2

Consequently,

The Steklov problem

In 1895 V. A. Steklov posed the following problem:

Let M be a smooth manifold of dimension ≥ 2 with smoothboundary bd(M). Find functions u on M and scalars σ ∈ R thatsatisfy,

∆u = 0 in M

∂νu = σu on bd(M)

The Steklov spectrum of M is set of all such σ.

The Steklov problem

In 1895 V. A. Steklov posed the following problem:

Let M be a smooth manifold of dimension ≥ 2 with smoothboundary bd(M). Find functions u on M and scalars σ ∈ R thatsatisfy,

∆u = 0 in M

∂νu = σu on bd(M)

The Steklov spectrum of M is set of all such σ.

The Steklov problem, rephrased

The Steklov spectrum is the spectrum of the“Dirichlet-to-Neumann” operator,

D : C∞(bd(M))→ C∞(bd(M)),

defined as follows.

Take u ∈ C∞(bd(M)).

Let u be the harmonic extension of u to M.

D(u) = (∂ν u)|bd(M).

Operator D is an elliptic pseudodifferential operator of order onewith the same principal symbol as

√∆bd(M).

Applied context: Electrical impedance tomography

In electrical impedance tomography electrical stimuli are placedaround a body. Resulting surface voltages are recorded and used toinfer the structure of objects inside the body.

This applied (numerical) technique relies on use of theDirichlet-to-Neumann operator.

Algebraic and Differential Geometry.Computational Algebra.

Teresa Arias-Marco

Department of MathematicsUniversity of Extremadura, Badajoz, Spain

January 14th, 2016

First Joint Meeting Evora-Extremaduraon Mathematics

algebraic and differential geometry. computational...

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