Riemannian manifolds

Sections in this chapter:

Introduction

The metric tensor

Volume form and Hodge star operator

Chistoffel symbols

Covariant derivative

Riemann and Ricci tensors

Submanifolds

Functions that stores the result

For other chapters go to Index

Introduction

In this chapter, we discuss the functions related with the metric tensor. The essential elements in a riemannian manifold are the metric and the connection and we will consider only riemannian connections. Moreover, throughout this chapter, only coordinate basis will be used. The generalization to the case of noncoordinate basis can be viewed in Noncoordinate basis.

The metric tensor

As we have seen in Coordinate systems and natural basis , more than one coordinate system is allowed in a session. Besides this, given a coordinate system or, better, a natural basis, several metrics can be defined, so, in order to specify a metric, two arguments are needed: the name of the metric and the basis in which it is expanded.
Associated to the default coordinate system XX, there is a default metric: the Euclidean metric. This metric is automatically adjusted to the dimension of the space. MetricNames shows the metrics defined at the moment and Metric returns the expression of the metric.

In[1]:= MetricNames

Out[1]= {Euclidean, XX, {X1, X2, X3}}

In[2]:= Metric[Euclidean, XX]

Out[2]= dX1*.dX1 + dX2*.dX2 + dX3*.dX3

In[3]:= InputCoordinates[XX, 4]

Out[3]= {XX, {X1, X2, X3, X4}}

In[4]:= Metric[Euclidean, XX]

Out[4]= dX1*.dX1 + dX2*.dX2 + dX3*.dX3 + dX4*.dX4

We define the spherical coordinates

and the short symbol e in order to make the input of the basis elements easier. Then, we declare the euclidean metric in the coordinate basis sph with InputMetric.

In[5]:= InputCoordinates[sph, {r, th, ph}]; e = ZZ[sph];

In[6]:= InputMetric[euclidean, sph,
            e[r, r] + r^2 e[th, th] + r^2 Sin[th]^2 e[ph, ph] ]

                  2             2        2
Out[6]= dr*.dr + r  dth*.dth + r  Sin[th]  dph*.dph

In[7]:= MetricNames

Out[7]= {Euclidean, XX, {X1, X2, X3}}
        {euclidean, sph, {r, th, ph}}

In[8]:= g = Metric[euclidean, sph]

                  2             2        2
Out[8]= dr*.dr + r  dth*.dth + r  Sin[th]  dph*.dph

Given a 2-covariant metric tensor

its inverse is the 2-contravariant tensor

being the inverse matrix of . InverseMetric and MetricDet are the functions that compute the inverse metric and the determinant of .

In[9]:= InverseMetric[euclidean, sph]

               2
        Csc[th]              -2
Out[9]= -------- Dph*.Dph + r   Dth*.Dth + Dr*.Dr
            2
           r
In[10]:= MetricDet[euclidean, sph]

          4        2
Out[10]= r  Sin[th]

ScalarProduct computes the scalar product of two vectors or 1-forms and Unitary gives the normalized version of a vector or form. Next, we obtain the norm of the vector

and then we make it unitary.

In[11]:= ScalarProduct[euclidean][e[-3], e[-3]]

          2        2
Out[11]= r  Sin[th]

In[12]:= Unitary[euclidean][e[-3]] // PowerExpand

         Csc[th]
Out[12]= ------- Dph
            r

Volume form and Hodge star operator

In a n-dimensional riemannian manifold the volume element

is the n-form

with tensor components

By using the volume element it is possible to define an isomorphism relating the k-forms and the (n-k)-forms. This task is performed by the Hodge star operator , that when acts upon the basis elements of the k-forms gives:

Then, if is a k-form, the components of (viewed as a tensor) are:

In physics, the result of applying the Hodge star operator to a tensor is known as the dual of a tensor and their components are given by the above expression.
The inverse operator is
The corresponding functions in TTC are VolumeForm, Hodge and InverseHodge. On the other hand TTC does not assume any default sign for the signature of the metric. You can use InputMetricSign[g,b,1 (-1)] if you want to avoid TTC ask for it

In[2]:= InputCoordinates[sph, {r, th, ph}]; e = ZZ[sph];:

In[3]:= g = e[1, 1] + r^2 e[2, 2] + r^2 Sin[th]^2 e[3, 3];

In[4]:= InputMetric[euclidean, sph, g];

In[5]:= w = r e[r] - r Sin[th] e[ph]

Out[5]= r dr - r Sin[th] dph

In[6]:= wH = Hodge[euclidean][w] // PowerExpand

Enter the sign of MetricDet[euclidean, sph],
  Interrupt[ ] or Dialog[ ]: +1

                      3
Out[6]= -r dr*^dth + r  Sin[th] dth*^dph

In[7]:= % // ExteriorToOuter

                                  3
Out[7]= -r dr*.dth + r dth*.dr + r  Sin[th] dth*.dph -

      3
>    r  Sin[th] dph*.dth

In[8]:= VolumeForm[euclidean, sph] // PowerExpand

         2
Out[8]= r  Sin[th] dr*^dth*^dph

In[9]:= InverseHodge[euclidean][%] // PowerExpand

Out[9]= 1

Christoffel symbols

The connection coefficients in a riemannian manifold when using a coordinate basis are often called the Christoffel symbols. Those of first kind are:

and those of the second kind are obtained from the previous ones by

The function Christoffel can compute both of them.
Here, we declare the polar coordinates in the plane and the euclidean metric written in these coordinates and then compute the second kind Christoffel symbol

In[2]:= InputCoordinates[polar, {r, phi}]; e = ZZ[polar];

In[3]:= g =  e[1, 1] + r^2 e[2, 2];

In[4]:= InputMetric[euclidean, polar, g]

                  2
Out[4]= dr*.dr + r  dphi*.dphi

In[5]:= Christoffel[euclidean, polar][1, -2, -2]

Out[5]= -r

We can also use the coordinate symbols with the corresponding signs instead

of the numerical indices in Christoffel. Next, we compute the first kind Christoffel symbol . In fact the first time we use Christoffel over a metric/coordinates the whole of Christoffel symbols are computed. See Functions that stores the result. For this the time for the first call of Christoffel can be very diferent than other calls

In[20]:= Christoffel[euclidean, polar][-r, -phi, -phi]

Out[20]= -r

ShowChristoffel yields a list of all the basic connection coefficients. Those with zero value and those related by the index symmetry

do not appear in the list.

In[21]:= ShowChristoffel[euclidean, polar]

Out[21]=  1
         K     = -r
           2 2

          2      1
         K     = -
           1 2   r

Covariant derivative

The covariant derivative of a tensor T of kind (p,q) is another tensor of kind (p,q+1) designated by the symbol

and with components (when p=q=1)

The covariant derivative receives different names in the bibliography as absolute derivative (Choquet pp.301) and gradient ( Misner pp.208). TTC implements this operator with AbsoluteD.

In an analogous way, the covariant derivative of the tensor T of kind (p,q) in the direction of the vector u is another tensor of kind (p,q) designated by and related to by

that is to say,

The implementation of this operator is CovariantD.
Next, we take polar coordinates in the plane and compute the covariant (or absolute) derivative of the vector obtaining as a result and later the covariant derivative of the vector in the direction of obtaining

In[2]:= InputCoordinates[polar, {r, phi}]; e = ZZ[polar];

In[3]:= g =  e[1, 1] + r^2 e[2, 2];

In[4]:= InputMetric[euclidean, polar, g]

In[5]:= AbsoluteD[euclidean][e[-1]]

        1
Out[5]= - Dphi*.dphi
        r

In[6]:= CovariantD[euclidean][e[-1]][ e[-1] + r e[-2] ]

Out[6]= 2 Dphi

Riemann and Ricci tensors

The Riemann tensor

corresponding to the metric

can be obtained from the Christoffel symbols by

Because of the symmetries it is usually better to work with the 4--covariant tensor . The corresponding TTC operator is Riemann.
From the Riemann tensor one can compute the Ricci tensor with Ricci and the curvature scalar with Curvature.
Let's see an example of computing Riemann and related objects. We declare the spherical coordinates {th, ph} with name sph2 and the metric SPH for a 2-sphere of radius a and compute , ,and R

In[2]:= InputCoordinates[sph2, {th, ph}]; e = ZZ[sph2];

In[3]:= InputMetric[SPH, sph2,
                a^2 (e[1, 1] + Sin[th]^2 e[2, 2]) ];
In[4]:= Riemann[SPH, sph2]

         2        2
Out[4]= a  Sin[th]  dth*.dph*.dth*.dph -

      2        2
>    a  Sin[th]  dth*.dph*.dph*.dth -

      2        2
>    a  Sin[th]  dph*.dth*.dth*.dph +

      2        2
>    a  Sin[th]  dph*.dth*.dph*.dth

In[5]:= Ricci[SPH, sph2]

                          2
Out[5]= dth*.dth + Sin[th]  dph*.dph

In[6]:= Curvature[SPH, sph2]

        2
Out[6]= --
         2
        a

Submanifolds

Let M be a riemannian manifold of dimension n with local coordinates xⁱ and

a submanifold of dimension m with coordinates x'^a .
If g is the metric tensor of M, the first fundamental form is the tensor in S:

. In components

The m tangent vectors in are .
When the dimension of S is m=n-1 then the normal form n can be written ( not unitary)

If N is the dual unitary vector obtained from n, the second fundamental form k is the 2--covariant tensor with components

The TTC functions used to compute these objects are TangentVector, NormalForm, FirstFundamentalForm and SecondFundamentalForm. The surface has to be declared using InputCoordinateChange.

Next, we define the surface of a torus in E³. One has to declare the coordinates of both spaces and the map (u,v)(x,y,z).

In[1]:= InputCoordinates[cart3, {x, y, z}];

In[2]:= InputCoordinates[param, {u, v}];

In[3]:= e = ZZ[cart3,];

In[4]:= InputMetric[Euclidean, cart3,
                e[1, 1] + e[2, 2] + e[3, 3] ];

In[5]:= InputCoordinateChange[ cart3, param,
               { x -> (a + b Cos[u]) Cos[v],
                 y -> (a + b Cos[u]) Sin[v],
                 z -> b Sin[u] } ]

Out[5]= {x -> (a + b Cos[u]) Cos[v],

>    y -> (a + b Cos[u]) Sin[v], z -> b Sin[u]}

We obtain the tangent vectors to the surface by using TangentVector.

The last argument indicates the number of the vector (which is negative since it is a subindex).

In[6]:= TangentVector[cart3, param][-1]

Out[6]= b Cos[u] Dz - b Sin[u] Sin[v] Dy - b Cos[v] Sin[u] Dx

In[7]:= TangentVector[cart3, param][-2]

Out[7]= (a + b Cos[u]) Cos[v] Dy - (a + b Cos[u]) Sin[v] Dx

In[8]:= n = NormalForm[Euclidean, cart3, param]

Out[8]= -(b Cos[u] (a + b Cos[u]) Cos[v]) dx -

>    b Cos[u] (a + b Cos[u]) Sin[v] dy -

>    b (a + b Cos[u]) Sin[u] dz

Finally, we compute the first and second fundamental forms of the torus.

In[39]:= FirstFundamentalForm[Euclidean, cart3, param]

          2                        2
Out[39]= b  du*.du + (a + b Cos[u])  dv*.dv

In[40]:= SecondFundamentalForm[Euclidean, cart3, param] // PowerExpand

Enter the sign 1 or -1 of the modulus of the
NormalForm[Euclidean, cart3, param], Interrupt[ ] or Dialog[ ]: 1

Out[40]= b du*.du + Cos[u] (a + b Cos[u]) dv*.dv

Functions that stores the result

A very common task in differential geometry is to compute the Riemann and Ricci tensors from a given metric

. It is usually computed first the inverse matrix

and then the Christoffel symbols

, the Riemann tensor

, the Ricci tensor

and the curvature R.

Through this process, is used in several places. To avoid this repeated use of InverseMetric, TTC stores the values it has found so that the computation is done efectively only the first time it is called.

There are other functions like InverseMetric that store values they have found::

Christoffel JacobianMatrix Ricci Curvature MetricDet SecondFundamentalForm FirstFundamentalForm NormalForm StructureCoefficient InverseMetric Riemann VolumeForm

For more details about the way TTC stores things and the way you can benefit of it see Big calculus

This page is maintained by XavierJaén.