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Inspired by differential equation $$y'(1+y^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$\nabla^{ (1+f^2)\nabla f}=0$$

Namely the vector field $(1+f^2)\nabla f$ is a parallel vector field. Every solution of this equation is called a cycloid function

On the other hand the classical cycloid has some singularities(non smooth points).

This motivates the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?

Inspired by differential equation $$y(1+y'^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$f(1+|\nabla f|^2)=c$$

On the other hand the classical cycloid has some singularities(non smooth points).

This motivates the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?

Inspired by differential equation $$y'(1+y^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$\nabla^{ (1+f^2)\nabla f}=0$$

Namely the vector field $(1+f^2)\nabla f$ is a parallel vector field. Every solution of this equation is called a cycloid function

On the other hand the classical cycloid has some singularities(non smooth points).

This motivate the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?

Inspired by differential equation $$y'(1+y^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$\nabla^{ (1+f^2)\nabla f}=0$$

Namely the vector field $(1+f^2)\nabla f$ is a parallel vector field. Every solution of this equation is called a cycloid function

On the other hand the classical cycloid has some singularities(non smooth points).

This motivates the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?

Inspired by differential equation $$y'(1+y^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$\nabla^{ (1+f^2)\nabla f}=0$$

Namely the vector field $(1+f^2)\nabla f$ would be a parallel vector field. Every solution of this equation is called a cycloid function

On the other hand the classical cycloid has some singularities(non smooth points).

This motivate the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?

Inspired by differential equation $$y'(1+y^2)=c$$

which generates the cycloid we consider the following differential equation on a Riemannian manifold:

$$\nabla^{ (1+f^2)\nabla f}=0$$

Namely the vector field $(1+f^2)\nabla f$ is a parallel vector field. Every solution of this equation is called a cycloid function

On the other hand the classical cycloid has some singularities(non smooth points).

This motivate the following question:

Is there an example of a Riemannian manifold which admits a globally smooth non constant cycloid function?