# Completion of Čech's and Kauchy's Results on the analysis of surfaces on which a canonical line passes through a fixed point

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## Summary

E. Čech's and J. Kaucky's[3] results on the analysis of surfaces on which a canonical line passes through a fixed point are completed by demonstrating the following theorems:

1. 1)

To every first-type solution, namely, the solutions β u 3 v 3 which satisfy the differential equation(B) of this paper, correspond 1 projective rotation surfaces mutually projectively applicable.

2. 2)

A projective rotation surfaces may allow at least one canonical line which passes through a fixed point. This yields a new property of these surfaces and completes the theorem of W. Lingenberg[9].

3. 3)

A projective rotation surface can admit at most two canonical lines passing through respective fixed points.

4. 4)

On surfaces corresponding to second-type solutions, namely β v 3 ≠ β u 3 and (k2 − 1)(k+3) ≠ 0, no canonical line passes through a fixed point, in contradiction to Kaucky's result[3].

5. 5)

With the exception of the surface denoted in this paper by (R), on surfaces which admit 2 transformation into themselves two canonical lines pass through respective fixed points.

6. 6)

On the surfaces denoted here by (R), Green's canonical line passes through a fixed point; it is the only surface which satisfies equation(B) and the system(C 1) and(C 2) with this property[11].

## Bibliography

1. [1]

G. Fubini -E. Čech,Geometria proiettiva differenziale, t. 1, Zanichelli, Bologna, 1926.

2. [1']

G. Fubini -E. Čech,Introduction à la géomètrie projective différentielle des surfaces, Paris, Gauthier-Villars, 1931.

3. [2]

E. Čech,Sur les surfaces dont toutes les courbes de Segre sont planes, Publication de la Faculté des Sci. de l'Université Masaryk, no. 11 (1922).

4. [3]

J. Kaucky,Etude de surfaces dont une droite canonique passe par un point fixe, ibidem, no. 109 (1929).

5. [4]

F. Marcus,Sur les reseaux de Koenigs, Revue de Math. pures et appliquées, t. 2 (Bucarest, 1957), pp. 555–559.

6. [5]

F. Marcus,Sur les surfaces de rotations projectives, Bull. de la classe des Sci. de l'Acad. Roy. de Belgique, 5e série,58 (1972–73), pp. 862–871.

7. [6]

Buchin Su,On a certain class of surfaces whose Darboux curves of one system are conics, The Tôhoku Math. Journal,36 (1933), pp. 241–252.

8. [7]

G. Fubini,Fondamenti di geometria proiettivo-differenziale, Rendiconti Circolo Mat. di Palermo, t. 43 (1919), pp. 1–46.

9. [7']

E. Bompiani,Rappresentazioni geodetico-proiettive fra due superficie, Annali di Mat. pura ed applicata, serie IV, t. 3 (1926), pp. 171–188.

10. [8]

F. Marcus,Quelques contributions à l'étude des deformations infinitésimales d'une surface en elle-même, Studü si Cercetari Stünţifice, Matematica Anul.,12 (1961), pp. 69–94.

11. [9]

W. Lingenberg,Zur Differentialgeometrie der Flächen die eine eingliedrige projektive Gruppe in sich gestatent etc., Math. Zts.,66 (1957), pp. 409–446.

12. [10]

F. Marcus,Sur les surfaces qui admettent ∞ 2 transformations projectives en elles-mêmes, Accad. Naz. dei XL, serie IV,24–25 (1974).

13. [11]

F. Marcus,On the results of J. Kaucky concerning the problem of the determination of surfaces for which a canonical line passes through a fixed point, in course of publication.

14. [12]

O. Boruvka,Sur certain types of surfaces qui sont projectivement applicables sur ellesmêmes, Publication de la Fac. des Sci. de l'Université Masaryk, Brno, 1924.

## Author information

Dedicated to ProfessorB. Segre

Entrata in Redazione il 27 luglio 1976.

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