It is a great mystery as to why there are so many excellent Russian born geometers/topologists (S. Novikov, M. Kontsevich, V. Voevodsky, G. Perelman, M. Gromov, Yu. Manin, I. Shafarevich, D. Fuchs, M. Postnikov) and so few good textbooks in Russian on those subjects.
So let me list the textbooks that are (in my opinion) still on par with the best modern textbooks:
- M. M. Postnikov, "Lectures in geometry", consisting of 6 textbooks:
- Semester 1, Analytic Geometry
- Semester 2, Linear Algebra
- Semester 3, Smooth Manifolds
- Semester 4, Differential Geometry
- Semester 5 (1), Riemann Geometry
- Semester 5 (2), Lie Groups
These textbooks are a slow-paced introduction to modern geometry, containing lots of optional material beyond standard facts. Sadly, these books are very much undervalued.
Especially good are Semesters 2-4, and 5(2).
The Linear Algebra textbook has lots of geometrical chapters (as to be expected from the title) about affine and projective geometries, quadrics and cones, etc.
Books on smooth manifolds and differential geometry are comparable to Spivak's 5 volume set and John Lee's "Introduction to smooth manifolds"; they also contain quite a lot of topological chapters on covering spaces, fundamental group, fiber bundles.
I think that most of these books are translated into English, but since you can read Russian there is no need to worry.
M. Postnikov also wrote textbooks on homotopy theory "Лекции по алгебраической топологии. Основы теории гомотопий (1984)" and "Лекции по алгебраической топологии. Теория гомотопий клеточных пространств (1985)", but I am not very familiar with these textbooks. From what I've heard they are considered quite good; they didn't get popularity because standard courses in algebraic topology rarely start with homotopy theory, and for someone who already saw homology/cohomology they might appear too wordy (~800 pages for elementary homotopy).
A. Kostrikin and Yu. Manin "Linear Algebra and Geometry" - considered one of the hardest textbooks on the subject (at the undergraduate level), since the authors try to avoid using coordinates and do everything in the abstract setting. The book uses a bit of category theory and has an extensive chapter on multilinear algebra.
S. Gelfand and Yu. Manin, "Methods of homological algebra" - one of the richest sources on homological (and even homotopical) algebra. The core of the book is chapter III where authors give a conceptual account of derived categories. [While not falling into geometry/topology category, homological algebra is one most indispensable tool in those fields, so I'll put it here]. It is interesting, that this was supposed to be only the first volume out of two, which to this day keeps me wondering what they had planned for the second volume (this was 1988, so the first volume was pretty much the state of the art. All of the $\infty$ techniques were years away!).
I. Shafarevich "Basic Algebraic Geometry 1: Varieties in Projective Space" and "Basic Algebraic Geometry 2: Schemes and Complex Manifolds" - this book is still recommended as one of the best introductions to algebraic geometry on undergraduate/first-year-graduate level by top universities.
A. Fomenko, D. Fuchs and V. Gutenmacher "Homotopic topology" - this book is the bible of Russian algebraic topologists (M. Khovanov still uses it for a course in Columbia). It has lots of material, but is rather terse and overall a less pleasant read than Hatcher. But you use it as a second textbook after Hatcher, where in one place you can find spectral sequences, cohomology operations, glimpses of K-theory and cobordism.
If you are interested in basics of classical, affine and projective geometries based on modern treatment of linear algebra you can check lecture notes of A. Gorodentsev. After reading these notes you would have a very good understanding of the structure of classical geometry. The content is somewhat similar to the textbook of M. Audin, but the exposition is better in my opinion.
You could also browse his excellent textbook on algebra which has lots of geometric material too.
- D. Burago, Yu. Burago, S. Ivanov, "A course in metric geometry" - a comprehensive exposition of a metric approach to geometry advanced by M. Gromov.
The above listed textbooks are best of the best of Russian textbooks on geometry and topology.
What follows are the textbooks that are good in some respects, but have significant drawbacks in my opinion.
S. Novikov and I. Taimanov "Modern Geometric Structures and Fields" - the book is about manifolds, differential geometry and topology, algebraic topology and application of all this machinery to theoretical physics. For my tastes the book uses coordinate arguments too much, probably trying to avoid abstract methods which could scare a physics-majored reader.
A. Mishchenko and A. Fomenko "A course of differential geometry and topology" - a bit dated exposition of differential geometry but very pictorial and intuitive (as noted by Javier Álvarez in his excellent post on smooth manifolds).
Yu. Manin "Введение в теорию схем и квантовые группы" - the book is an introduction into algebraic geometry using schemes and quantum groups in noncommutative geometry. I have used it on several occasions, but the exposition is very terse and a bit unmotivated.
To make the survey of textbooks complete I would like to mention the textbooks by
- V. Zorich "Mathematical analysis I" and "Mathematical Analysis II" - which are not on geometry/topology, however the second volume has extensive exposition of analysis on manifolds. Moreover the exposition of classical analysis uses ideas from general topology, classic geometry and linear algebra. I think it is the best analysis textbook to date.
- A. Pirkovskii's lecture notes on Functional Analysis - a very friendly modern introduction to functional analysis. My favorite on functional analysis
- E. Vinberg "A course in algebra" - a good algebra textbook similar to Artin's textbook, but in my opinion the aforementioned textbook of Gorodentsev is better. I list it here because Vinberg is translated into English.
- V. Arnold "Ordinary Differential Equations" and "Mathematical Methods of Classical Mechanics" are considered to be somewhat ambitious non-standard introductions into differential equations and dynamical systems, but I don't like the exposition that much.
I think these were the best Russian textbooks (known to me) that are still good to learn from. There are also good Russian textbooks in probability theory, measure theory, combinatorics, etc, but they are either dated or I didn't have a chance to browse through them.
!!! There are also expository monographs (just under 100 volumes!) similar to Bourbaki volumes called "VINITI. Contemporary problems in mathematics. Fundamental branches". These books were written by the best Soviet mathematicians: Dynamical systems by V. Arnold, Differential Topology by S. Novikov, etc. Lots of gems in those books, but describing their content would take a whole separate long post. Such a shame these books didn't get popular (as they surely deserved to had they not been published right around time of the collapse of the USSR). Very briefly, as far as I know the most widely used in geometry are:
- Dynamical systems 4, "Symplectic geometry" by Arnold and Givental
- Algebra 1, "Basic concepts of algebra", by Shafarevich
- Lie groups and Lie algebras 1, "Foundations of the theory of Lie groups", by Vinberg and Onishchick
- Algebraic geometry 1, by Shokurov and Danilov
- Algebraic geometry 2, by Danilov (a very popular "Cohomology of algebraic varieties"), Iskovskikh, Shafarevich
- Geometry 1, "Basic ideas and concepts of differential geometry", by
Alekseevskii, Vinogradov, Lychagin
- A unique in its scope recent account of the state of complex-analytic geometry in "Complex analysis - Several variables", volumes 1-7
This effort was later picked up by Springer in their "Encyclopaedia of Mathematical Sciences", where many of the above volumes found their English translation and the series continues with monographs from mathematicians from all over the globe.
Hope that helps.