In this study, we examine the possible relations between the Frenet planes of any given two curves in three dimensional Lie groups with left invariant metrics. We explain these possible relations in nine cases and then introduce the conditions that must be met to coincide with the planes of these curves in nine theorems.

The theory of curves has an important role in differential geometry studies. In the theory of curves, one of the interesting problems is to investigate the relations between two curves. The Frenet elements of the curves have an effective role in the solution of the problem.

For example, if the principal normal vectors coincide at the corresponding points of the curves

In a three-dimensional Lie group

Karakuş et al. have examined the possibility of whether any Frenet plane of a given space curve in a three-dimensional Euclidean space is also any Frenet plane of another space curve in the same space [

In this study, we examine the possible relations between the Frenet planes of given two curves in three dimensional Lie groups with left invariant metrics.

Let

If the group is unimodular, we have a (positively oriented) orthonormal frame of left-invariant vector fields

where

If the group is nonunimodular, we have an othonormal frame

see [

Using the Koszul formula the covariant derivatives

for unimodular and nonunimodular cases, respectively.

The cross-products of the vectors

We have

Let

Let

Along the curve

The Frenet and the dot-Frenet frames are connected by

Define a group-curvature

(see [

Let

So we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By using

By using the

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

Thus, we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have the equation

If we multiply the

Thus we introduce the following theorem:

Thus, we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By

By using the

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

So we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By the

By using

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

Thus, we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have the equation

If we multiply the

This means that,

Thus we introduce the following theorem:

Thus, we have following relations between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By the

By using

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

So we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By the

By using

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

Thus, we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have the equation

If we multiply the

Thus we introduce the following theorem:

Thus, we have following relation between the curves

Calculating the dot-derivative of the

We know that

And so, we have

By the

By using

Calculating the dot-derivative of the

If we multiply the

By using

Thus we introduce the following theorem:

It is well known that every smooth curves have a moving Frenet frame. This paper examines the relations between Frenet planes of two smooth curves in three dimensional Lie groups with left-invariant metric. There are nine possible relations that can occur. For each cases, we give conditions by nine theorems as above. These results are generalizations for relations between Frenet planes of two curves in three dimensional Euclidean spaces. By the paper's results, one will be able to investigate of special curve couples in three-dimensional Lie groups with left-invariant metric and correlate their results.