Multidisciplinary Journal of Science and Technology

SUBMERSION IN EUCLIDIAN SPACE

Salimova Gulhayo

Keywords: Yevklid fazosi, differensiallanuvchi akslantirish, botirish, submersiya, Yakobi matritsasi, sath sirtlari, regulyar sirt, diffeomorfizm, Riman submersiyasi, ortogonal proyeksiya.

Abstract

This article deeply studies submersions, one of the important classes of differentiable reflections defined in Euclidean space. The concepts of submersion, immersion, and diffeomorphism are clearly defined and the connections between them are analyzed. The main properties of submersions, in particular, the fundamental theorem that their preimages form differentiable surfaces, are highlighted. The geometric meaning of submersions is shown using the example of plane surfaces, and classical surfaces such as parabolas, elliptic, and hyperbolic paraboloids are analyzed. The concept of Riemann submersion is also introduced and its main properties are described using the example of orthogonal projection. The theoretical results in the article are supported by concrete examples and are of great importance in the study of differential geometry and Riemannian geometry.

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