Differential Geometry


Humboldt-University zu Berlin


Campus Adlershof

At the Humboldt-University zu Berlin, I specialized in differential geometry, in particular semi-Riemannian geometry, and Lie theory. The concepts are used in physics to fomulate:

In 2006, I submitted my diploma thesis on Lorentzian Ricci-flat homogeneous manifolds.

On Lorentzian Ricci-flat Homogeneous Manifolds
Ricci-flat Invariant Geometry in Four Dimensions
Raytracing Special Relativity
Biharmonic Distance
Spin Transformations

Tensor fields are the essence of differential geometry. Relevant computations, such as the determination of the Riemannian curvature, usually are not doable by hand. All this is covered in our differential geometry package for Mathematica 5.2. We also provide operations related to Lie algebras Lie and Lie groups.

Differential Geometry with Mathematica
Latest version of the (DG)kernel notebook
Documentation
Tensors
Manifolds
Lie-algebras
Lie-groups
The beauty of Nature lies in detail; the message in generality.
Optimal appreciation demands both and I know no better tactic than
the illustration of exciting principles by well-chosen particulars.
Stephen Jay Gould

On Lorentzian Ricci-Flat Homogeneous Manifolds


two parametrizations of moduli spaces of 4-dimensional Lorentzian homogeneous
triples with isotropy dim h=1, and curvature Ric=0

The chapter "Introduction" from my thesis:

A homogeneous space is the coset manifold G/H, where G is a Lie group, and H is a closed Lie subgroup of G. The canonic mappings related to a homogeneous space are the projection, and the left-action. The differential of the left-action extends tensors of special form to invariant tensor fields on the manifold G/H. For instance, an invariant metric on G/H originates from a single scalar product B. A homogeneous triple associated to the homogeneous space G/H with invariant metric is (g,h,B), where g is the Lie algebra of G, h<g is the subalgebra induced by the subgroup H, and B is the scalar product on a vector space complement m with g=h+m induced by the metric.

For a homogeneous space with invariant metric, geometric notions such as the Levi-Civita connection, and the curvature can be derived at a single point of G/H in terms of the homogeneous triple. The curvature tensor obtained this way translates to the invariant curvature tensor field on the semi-Riemannian manifold G/H. Locally, the homogeneous triple uniquely determines the corresponding semi-Riemannian homogeneous space. In the thesis, we are interested in homogeneous triples that relate to homogeneous spaces with the following geometric properties:

Metrics of index 1 are the core in relativity theory. From the geometric viewpoint, Riemannian-flat homogeneous spaces are not particularly interesting. A Ricci-flat homogeneous space is naturally an Einstein manifold.

On Lorentzian Ricci-Flat Homogeneous Manifolds (Thesis) * lorentzian_ricci... 630 kB
Details to Example 3.7 thes_exm37 link
* submitted on the 30th Sep 2006 as Diplomarbeit

The major contributions are summarized on the last page of the thesis. The picture above visualizes two parametrizations of moduli spaces of 4-dimensional Lorentzian homogeneous triples with isotropy dim h=1, and curvature Ric=0.

And the end of all our exploring
will be to arrive where we started
and know the place for the first time.
Thomas Stearns Eliot

Subsequently, two research papers have cited the thesis:

Alan Coley, Sigbjørn Hervik, Nicos Pelavas: Lorentzian spacetimes with constant curvature invariants in four dimensions, Classical and Quantum Gravity 26, May 2009

Abstract: In this paper we investigate four-dimensional Lorentzian spacetimes with constant curvature invariants (CSI spacetimes). We prove that if a four-dimensional spacetime is CSI, then either the spacetime is locally homogeneous or the spacetime is a Kundt spacetime for which there exists a frame such that the positive boost weight components of all curvature tensors vanish and the boost weight zero components are all constant. We discuss some of the properties of the Kundt-CSI spacetimes and their applications. In particular, we discuss I-symmetric spaces and degenerate Kundt-CSI spacetimes.

Mohamed Boucetta: Ricci flat left invariant pseudo-Riemannian metrics on 2-step nilpotent Lie groups, arXiv:0910.2563v1, Oct 2009

Abstract: We determine all Ricci flat left invariant pseudo-Riemannian metrics of signature (q,q+p) on 2-step nilpotent Lie groups, where q=1 or q=2. We show that the 2k+1-dimensional Heisenberg Lie group H2k+1 carries a Ricci flat left invariant Lorentzian metric if and only if k=1. We show also that for any 2≤q≤k, H2k+1 carries a Ricci flat left invariant pseudo-Riemannian metric of signature (q,q+p).

If not now, then when?
If not me, then who?

Biharmonic Distance


Hook

Sphere

Beethoven bust

Yaron Lipman, Raif M. Rustamov, and Thomas A. Funkhouser introduced Biharmonic Distance at Siggraph 2011. Biharmonic distance is a construction of a metric on a continous or discrete (triangulated) surface. The resulting metric d(x,y), i.e. a notion of distance between points x and y, has very desirable properties. I quote from the paper:

Biharmonic Distance (Matlab) * biharmonic.zip 170 kB
* for a demo run biharmonicdemo.m in Matlab.

My implementation that is listed below works well for triangulated meshes with less than 3k vertices.

function dB=biharmonic(T,V)
% input: T (3xN) and V (3xM) so that
%  trimesh(T',V(1,:),V(2,:),V(3,:))
%  plots the triangular mesh
% output: dB (MxM) matrix where dB(i,j)
%  is distance between vertex i and j

nT=size(T,2);
nV=size(V,2);

AT=zeros(nT,1);
for c1=1:nT
  pnt=V(:,T(:,c1));
  AT(c1)=norm(cross(pnt(:,2)-pnt(:,1),pnt(:,3)-pnt(:,1)));
end

AV=zeros(nV,1);
ind={};
for c1=1:nT
  c3=T(:,c1);
  AV(c3)=AV(c3)+AT(c1);
  for c2=[c3 c3([2 3 1]) c3([3 1 2])]
    i=c2(1);
    j=c2(2);
    k=c2(3);
    v1=V(:,i)-V(:,k);
    v2=V(:,j)-V(:,k);
    fac=norm(v1)*norm(v2);
    if fac<eps
      fprintf('triangle %i is degenerate\n',c1)
    else
      val=cot( acos( dot(v1,v2) / fac ) );
      ind{end+1,1}=[
        i i  val
        i j -val
        j i -val
        j j  val
      ];
    end
  end
end
ind=cell2mat(ind);
Lc=sparse(ind(:,1),ind(:,2),ind(:,3),nV,nV);

gd=pinv(full(Lc*spdiags(6./AV,0,nV,nV)*Lc));
dia=diag(gd);
dB=zeros(nV);
for i=1:nV
  dB(:,i)=sqrt(gd(i,i)+dia-2*gd(:,i));
end

I have chosen the command pinv for simplicity and stability, as the Matlab command svds for sparse matrices does not neccessarily converge. Section 3.3 Practical Computation in the paper describes how to solve a series of systems of linear equations to yield the exact metric in a more efficient computation.

I suggested that we might compare earthquakes in terms
of the measured amplitudes recorded at these stations,
with an appropriate correction for distance.
Charles Francis Richter

Spin Transformations


input 1

output 1

input 2

output 2

Keenan Crane, Ulrich Pinkall, and Peter Schröder introduced Spin Transformations of Discrete Surfaces at Siggraph 2011. The input to their algorithm is a triangled mesh and a function f that maps each triangle to a real value. The output is a smoothly deformed mesh that has

Initially, I started to implement the method according to the derivation in their paper. But then, Keenan made his implementation open-source, so I simply ported his C++ code to Matlab. The examples that are displayed to the side are also taken from Keenans website. The colors of the triangles in the illustrations to the side represent the values of f.

Spin Transformations (Matlab) * spin.zip 790 kB
* for a demo run spindemo.m in Matlab.
function V=spin(T,V,rho)
% input: T (3xN) and V (3xM) so that
%  trimesh(T',V(1,:),V(2,:),V(3,:)) plots the triangular mesh,
%  rho (1xN) values of conformal scaling
% output: V (3xM) vertices of transformed mesh

nT=size(T,2);
nV=size(V,2);
plc=-3:0;

E=sparse(4*nV,4*nV);
edg=zeros(4,3);
for c1=1:nT
  tri=T(:,c1);
  pnt=V(:,tri);
  A=norm( cross( pnt(:,2)-pnt(:,1) , pnt(:,3)-pnt(:,1) ) )/2;
  a=-1/(4*A);
  b=rho(c1)/6;
  c=jiH([A*rho(c1)*rho(c1)/9 0 0 0]);
  for c2=1:3
    edg(:,c2)=[0;V(:, tri(mod(c2+1,3)+1) )-V(:, tri(mod(c2+0,3)+1) )];
  end  
  ini=[tri(1)*4+plc tri(2)*4+plc tri(3)*4+plc];
  E(ini,ini)=E(ini,ini) + [
    jiH(jiH(a*edg(:,1))*edg(:,1)) + c ...
    jiH(jiH(a*edg(:,1))*edg(:,2) + b*(edg(:,2)-edg(:,1)))+c ...
    jiH(jiH(a*edg(:,1))*edg(:,3) + b*(edg(:,3)-edg(:,1)))+c
    jiH(jiH(a*edg(:,2))*edg(:,1) + b*(edg(:,1)-edg(:,2)))+c ...
    jiH(jiH(a*edg(:,2))*edg(:,2)) + c ...
    jiH(jiH(a*edg(:,2))*edg(:,3) + b*(edg(:,3)-edg(:,2)))+c
    jiH(jiH(a*edg(:,3))*edg(:,1) + b*(edg(:,1)-edg(:,3)))+c ...
    jiH(jiH(a*edg(:,3))*edg(:,2) + b*(edg(:,2)-edg(:,3)))+c ...
    jiH(jiH(a*edg(:,3))*edg(:,3)) + c ];
  if ~mod(c1,500); fprintf('.'); end
end
fprintf('\n')

lam=zeros(4*nV,1);
lam(1:4:end)=1;  
for c1=1:11
  cnv=lam;
  lam=E\lam;
  lam=lam/norm(lam);
end
res=(E*lam)./lam;  
fprintf('mean %e, var %e, delta %e\n',mean(res),var(res),norm(cnv-lam))

L  =sparse(4*nV,4*nV);
ome=zeros(4*nV,1);
for c1=1:nT
  for c2=1:3
    k0=T(mod(c2-1,3)+1,c1);
    k1=T(mod(c2+0,3)+1,c1);
    k2=T(mod(c2+1,3)+1,c1);
    u1=V(:,k1)-V(:,k0);
    u2=V(:,k2)-V(:,k0);
    cta=dot(u1,u2) / norm( cross(u1,u2) );
    h=jiH([cta*0.5 0 0 0]);
    ini=[k1*4+plc  k2*4+plc];
    L(ini,ini)=L(ini,ini)+[ h -h;-h h];
    if k1>k2
      k3=k1; k1=k2; k2=k3; % swap
    end
    lm1=jiH(lam(k1*4+plc));
    lm2=jiH(lam(k2*4+plc));
    edv=jiH([0;V(:,k2)-V(:,k1)]);
    til=lm1'*edv*lm1/3 + lm1'*edv*lm2/6 + lm2'*edv*lm1/6 + lm2'*edv*lm2/3;
    ome(k1*4+plc,1)=ome(k1*4+plc,1)-cta*til(:,1)/2;
    ome(k2*4+plc,1)=ome(k2*4+plc,1)+cta*til(:,1)/2;
  end
  if ~mod(c1,500); fprintf('.'); end
end
fprintf('\n')

ome=reshape(ome,[4 nV]);
ome=ome-repmat(mean(ome,2),[1 nV]);
ome=reshape(ome,[4*nV 1]);
ome=L\ome;
ome=reshape(ome,[4 nV]);
ome=ome-repmat(mean(ome,2),[1 nV]);
nrm=sum(ome.*ome,1);
ome=ome/sqrt(max(nrm));
V=ome(2:end,:);

function h=jiH(pnt)
a=pnt(1); b=pnt(2); c=pnt(3); d=pnt(4);
h=[ a -b -c -d
    b  a -d  c
    c  d  a -b
    d -c  b  a];
Those who make peaceful revolution impossible
will make violent revolution inevitable.
John Fitzgerald Kennedy