By Yu.D. Burago, V.A. Zalgaller
The unique model of this text used to be written greater than fiveyears in the past with S. Z. Shefel',a profound and unique mathematician who died in 1984. Sincethen the geometry of surfaces has endured to be enriched with rules and effects. This has required alterations and additions, yet has now not prompted the nature of the object, the layout ofwhich originated with Shefel'. with out realizing to what volume Shefel' may have authorized the adjustments, I should still however wish to devote this text to his reminiscence. (Yu. D. Burago) we're attempting to country the qualitative questions of the idea of surfaces in Euclidean areas within the shape during which they seem to the authors at this time. This description doesn't totally correspond to the ancient improvement of the topic. the idea of surfaces used to be built within the first position frequently because the three concept of surfaces in third-dimensional Euclidean area E ; notwithstanding, it is sensible to start through contemplating surfaces F in Euclidean areas of any size n~ three. This method permits us, specifically, to install a brand new gentle a few three unsolved difficulties of this built (and with regards to surfaces in E rather whole) idea, and in lots of instances to consult the connections with the current level ofdevelopment of the speculation of multidimensional submanifolds. The prime query of the object is the matter of the relationship among sessions of metrics and sessions of surfaces in En.
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