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.

Writing

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

we have

,    ,    .

and

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

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

This occurs when
and    .

So, the elliptic points lie on these curves

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.

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
and    .

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

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).

REFERENCES:

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

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

In matrix form, the Schwarzschild metric can be written as

The inverse metric then becomes

The Christoffel symbols are calculated from the formula

where is the inverse matrix of .

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

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

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

where  is the proper time and  is the four-velocity. In this case, .

Thus,

3. The Riemann Curvature Tensor

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

The nonzero components are

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 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

yields zero.

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

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