Geodesic in plane when metric depends on single variable?

In summary, a p+1 dimensional manifold with a (fisher) metric that is a function of beta only, has flat submanifolds for a constant beta and straight line geodesics in these submanifolds. However, the geodesic connecting two points does not necessarily lie in a 2D-plane, unless the entire manifold is a plane. This realization could potentially improve the numerical determination of geodesics and geodesic distances between probability densities in this type of models.
  • #1
rhe
1
0
Hello,

I have a p+1 dimensional manifold describing the parameter space of a family of probability densities. The p+1 dimensions are (beta, t1, t2, ..., tp), all reals, and beta restricted to the positive reals. The (fisher) metric on this manifold is a function of beta only, hence the submanifolds for a constant beta are flat, and the geodesics are straight lines in these submanifolds.

Consider two points P=(beta1,0,0,...,0) and Q=(beta2,t1,t2,...,tp), and the geodesic connecting these two points. I suspect that this geodesic lies in the 2D-plane spanned by the vector (1,0,0,...,0) and the unit vector pointing towards Q. Is this true?

If so, this could significantly speed up the numerical determination of geodesics and geodesic distances between probability densities for this type of models.

Thank you
 
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  • #2
rhe said:
Consider two points P=(beta1,0,0,...,0) and Q=(beta2,t1,t2,...,tp), and the geodesic connecting these two points. I suspect that this geodesic lies in the 2D-plane spanned by the vector (1,0,0,...,0) and the unit vector pointing towards Q. Is this true?


No, it's not, unless your manifold is a plane.
 

1. What is a geodesic in the plane when the metric depends on a single variable?

A geodesic in the plane is a curve that follows the shortest path between two points, taking into account the changing metric of the plane as a function of a single variable. Think of it as the equivalent of a straight line in Euclidean space, but adjusted for the varying curvature of the plane.

2. How is the metric of the plane related to the geodesic?

The metric of the plane determines the distance between two points and the angle of intersection between two curves. This information is used to calculate the geodesic, which represents the shortest distance between two points on the plane with the given metric.

3. What is the significance of the metric depending on a single variable?

When the metric of the plane depends on a single variable, it means that the curvature of the plane is not constant and varies in a specific direction. This allows for a more accurate representation of the shortest path between two points, as it takes into account the changing curvature of the plane.

4. How is the geodesic calculated in this scenario?

The geodesic is calculated by using the Euler-Lagrange equation, which takes into account the metric of the plane and its derivatives. This equation helps to determine the curve that minimizes the distance between two points on the plane with the given metric.

5. Can a geodesic in the plane exist when the metric depends on multiple variables?

Yes, a geodesic can exist in a plane with a metric that depends on multiple variables. In this case, the geodesic is calculated using the partial differential equations of the metric, taking into account the multiple variables that affect its curvature. This can result in a more complex and accurate geodesic curve in the plane.

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