# A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction and Statements of Results

- $m>n,$
- $gcd\left(m,n\right)=1,$
- $m+n\equiv 1$ (mod 2) (see [1]).

“Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel”.

**Theorem**

**1.**

**Remark**

**1.**

## 2. Preliminaries

## 3. Proof of Theorem 1

**Case a.**$K=1$ and ${M}^{2}$ is totally geodesic. This case is not possible, already we discussed it above.

**Case b.**$K=0$ and ${M}^{2}$ is an open piece of the Clifford torus. Thus, ${K}_{ext}=-1$, which does not fulfill Equation $\left(7\right)$.

**Proof**

**of**

**Theorem.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Aydin, M.E.; Mihai, A.
A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms. *Mathematics* **2020**, *8*, 444.
https://doi.org/10.3390/math8030444

**AMA Style**

Aydin ME, Mihai A.
A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms. *Mathematics*. 2020; 8(3):444.
https://doi.org/10.3390/math8030444

**Chicago/Turabian Style**

Aydin, Muhittin Evren, and Adela Mihai.
2020. "A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms" *Mathematics* 8, no. 3: 444.
https://doi.org/10.3390/math8030444