Zorich's theorem

In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. ^[1] The result was conjectured by M. A. Lavrentev in 1938. ^[2]

Theorem

Every locally homeomorphic quasiregular mapping ${\displaystyle f:R^{n}\rightarrow R^{n}}$ for ${\displaystyle n\geq 3}$, is a homeomorphism of ${\displaystyle R^{n}}$. ^[3]

The fact that there is no such result for ${\displaystyle n=2}$ is easily shown using the exponential function. ^[4]

References

1. Zorič, V. A. (1967). "Homeomorphism of quasiconformal space maps". Proceedings of the USSR Academy of Sciences . 176: 31–34. MR   0223568. As cited by Zorich (1992)
2. Lavrentieff, M. (1938). "Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions". Proceedings of the USSR Academy of Sciences . 20: 241–242. As cited by Zorich (1992)
3. Zorich, Vladimir A. (1992). "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems". In Vuorinen, Matti (ed.). Quasiconformal Space Mappings: A collection of surveys 1960-1990. Lecture Notes in Mathematics. Vol. 1508. Germany: Springer-Verlag. pp. 132–148. doi:10.1007/BFB0094243. ISBN   3-540-55418-1. LCCN   92012192. OCLC   25675026. S2CID   116148715 . Retrieved February 10, 2024.
4. Zorich (1992), p. 135.