In this article, we shall describe elliptic, hyperbolic, parabolic points of a surface. We choose a torus as the surface for this purpose because it is a good example where these points occur in various parts of the surface.


Consider an elliptical torus:

The parametric form of an elliptical torus is given by

    torus[a,b,c](u,v)=((a + b cos v) cos u, (a + b cos v) sin u, c sin v),

where u,v ∈ [0, 2π).


Here is a graph of a torus with a=2, b=1, c=1:



By fixing the v-parameter, the following graph shows several u-coordinate curves on the torus[2,1,1]:



The Gaussian curvature of the torus can be computed using the equation


where E, F, G are the coefficients of the first fundamental form relative to the torus, and e, f, g are the coefficients of the second fundamental form.


         x(u,v)≡torus[a,b,c](u,v), and denote subscript as partial derivative,

we have 

       index_2.png,    index_3.png,    index_4.png.



where U is the unit normal vector at points on the torus.


It is interesting to note that the Gaussian curvature for the torus depends only on v



As for our example, consider a torus with a=2, b=1, c=1, where u,v ∈ [0,2 π). The Gaussian curvature for this torus becomes,



Plotting the Gaussian curvature with respect to v, we obtain

         curvature of torus


Points on the surface is called elliptic if the Gaussian curvature is positive, that is, for our case


This occurs when
        index_9.png    and    index_10.png.

So, the elliptic points lie on these curves

         elliptic points on torus


Points on the surface is called hyperbolic if the Gaussian curvature is negative, that is, for our case


This occurs when

The following graph shows hyperbolic points that lie on some of the curves.

         hyperbolic points on torus


The shape operator of the torus[2,1,1] can be computed and found to be nonzero



Points on the surface is called parabolic if the Gaussian curvature vanishes and the shape operator is nonzero.

This occurs when
        index_14.png    and    index_15.png.

There are only two curves that lie on the torus[2,1,1] which are parabolic as is shown in the following graph:

        parabolic points on torus

Finally the figure below shows curves on the torus[2,1,1] where points are elliptic (blue curves), hyperbolic (green curves), and parabolic (red curves).

        elliptic, hyperbolic, and parabolic points on torus


Gray, A., Abbena, E., Salamon, S. Modern Differential Geometry of Curves and Surfaces with Mathematica, 3nd ed. Boca Raton, FL: CRC Press, 2006.

Tan, S., Vector Calculus Using Mathematica, 1st ed. Lulu, 2018.

The Schwarzschild solution is the first exact solution to the Einstein Field Equations of General Relativity, which makes use of Schwarzschild coordinates and Schwarzschild metric.

The Schwarzschild metric describes a static and spherically symmetric gravitation field in the empty region of spacetime near a massive spherical object. It is an exact solution to the Einstein Field Equations for non-rotating spherical objects. However, it provides a good approximation to the gravitational field of a slowly rotating bodies such as the Sun or the Earth. The solution also leads to a derivation of the Schwarzschild radius which is the size of the event horizon of a non-rotating black hole.

In this article, we are going to give a summary of the Christoffel symbols, the geodesic equations, the Riemann curvature tensor, the Ricci tensor, Ricci scalar of curvature by using the Schwarzschild metric. Since these calculations are lengthy and tedious, we will not show the details here but will only display the results.

The Schwarzschild metric is

the schwarzschild metric
where M is the mass of gravitational source, G is the gravitational constant, and the schwarzschild coordinates are the Schwarzschild coordinates.

In matrix form, the Schwarzschild metric can be written as

the Schwarzschild metric matrix form

The inverse metric then becomes

the schwarzschild metric, inverse matrix form

The Christoffel symbols are calculated from the formula

the christoffel symbols

where inverse metricis the inverse matrix of metric.

Next, we will list the non-zero results of our calculations.


1. The Christoffel symbols – there are nine non-vanishing components

the christoffel symbols for the schwarzschild metric


2. The Geodesic equations

After obtaining the Christoffel symbols we can calculate the geodesic equations. The four geodesic equations allow physicists to describe how objects and photons move in Schwarzschild spacetime.

The four components of the geodesic equations are obtained by using the following basic equation

the geodesic equations

where tau is the proper time and four-velocity is the four-velocity. In this case, four-velocity in the schwarzschild coordinates.


four-velocity in the schwarzschild metric


3. The Riemann Curvature Tensor

The components of the Riemann tensor, the riemann tensor are calculated using the definition

the riemann tensor definition

The nonzero components are

the riemann tensor in schwarzschild metric

the riemann tensor in schwarzschild metric

Note:  the Riemann curvature tensor is anti-symmetric on the first and second pair of indices.


4. The Ricci tensor and Ricci scalar

The Ricci tensor can be formed by contracting the first and the third indices of the Riemann curvature tensor (which has the first index already raised).

the ricci tensor

The results of our calculation, the Ricci tensor has vanishing components. Likewise the Ricci scalar which is formed by using the inverse metric and the Ricci tensor
the ricci scalar
yields zero.

The Einstein tensor the einstein tensor is found from the tensors already calculated. We obtain

vacuum einstein equation

This means that the Schwarzschild metric satisfied the vacuum Einstein equation.