Question

Find the geodesics on the parabolic cylinder $y=x^2$.

Short answer

Geodesics largely follow u,z parameters curved in initial parametric input.

Step-by-step solution

Title - Parametrize the Surface

A good starting point is to parametrize the surface of the parabolic cylinder. Use the surface equation and a parameter. Let the parameters be: $x=u$ $y=u^2$ $z=v$

Title - Find the Metric Tensor

Next, compute the metric tensor. Start by finding the partial derivatives of the parametric equations with respect to the parameters $u$ and $v$:${\mathbf{r}}_u=(1,2u,0)$${\mathbf{r}}_v=(0,0,1)$The metric tensor $g_{ij}$ is then:$$g_{ij}=\left(\begin{array}{c}{\mathbf{r}}_u{\mathbf{r}}_v\end{array}\right)\left(\begin{array}{cc}{\mathbf{r}}_u^T& {\mathbf{r}}_v^T\end{array}\right)$$Computing the inner products:$$g_{11}={\mathbf{r}}_u\cdot {\mathbf{r}}_u=1+4u^2,g_{12}={\mathbf{r}}_u\cdot {\mathbf{r}}_v=0,g_{22}={\mathbf{r}}_v\cdot {\mathbf{r}}_v=1$$So, the metric tensor $g_{ij}$ is:$$g_{ij}=\left(\begin{array}{ccc}1+4u^2& 00& 1\end{array}\right)$$

Title - Write Down the Geodesic Equations

With the metric tensor known, write down the geodesic equations using the Christoffel symbols. The geodesic equations are:$$\frac{d^2{u}^k}{d{\tau }^2}+{\mathrm{\Gamma }}_{ij}^k\frac{d{u}^i}{d\tau }\frac{d{u}^j}{d\tau }=0$$where the Christoffel symbols ${\mathrm{\Gamma }}_{ij}^k$ are calculated from the metric tensor. For the metric tensor $g_{ij}$, Christoffel symbols can be computed, but in this specific case, it's easier to see the inherent symmetry and deduce these values directly.

Title - Solve the Geodesic Equations

Given the simplicity of the parameterization and metric, simpler geodesic paths occur when the complexity of curvature is bounded by parameters. This means lines parallel to the z-axis and slight curvatures due to $u$. General solutions are combinations of connections on lines projected on z from the initial lines along the parabolic x,y relationship. Numerical methods might be required for more detail.

Title - Interpret the Physical Meaning

Translate the mathematical solutions into a physical and visual interpretation. Normally, the shortest path on the cylinder will follow the curves of the parabolic surface but projected, such paths simplify for lines along 'z', thus direct translation happens more simplified.

Key concepts

Parametric Surfaces

To understand geodesics, we start with the concept of parametric surfaces. This is a way to describe a surface using parameters and equations. For the parabolic cylinder, the surface is described by the equation $y=x^2$. We use parameters $u$ and $v$ to express this surface as:

• $x=u$
• $y=u^2$
• $z=v$

This means any point on the surface can be described using these two variables. This method helps us simplify and break down complex surfaces into manageable pieces.

Metric Tensor

The metric tensor is a key concept in differential geometry as it helps measure distances on surfaces. Once we have our parametric equations, we find partial derivatives with respect to parameters ($u$ and $v$). For the parabolic cylinder:

• ${\mathbf{r}}_u=(1,2u,0)$
• ${\mathbf{r}}_v=(0,0,1)$

The metric tensor $g_{ij}$ is built using these partial derivatives. We compute inner products:

• $g_{11}={\mathbf{r}}_u\cdot {\mathbf{r}}_u=1+4u^2$
• $g_{12}={\mathbf{r}}_u\cdot {\mathbf{r}}_v=0$
• $g_{22}={\mathbf{r}}_v\cdot {\mathbf{r}}_v=1$