## Lectures in Geometry – Semester 2 Linear Algebra and Differential Geometry – Postnikov

In this post, we will see the book Lectures in Geometry – Semester 2 Linear Algebra and Differential Geometry by M. Postnikov. This book is the first one of a five part Lectures in Geometry series. So far we have volumes 1, 2 and 5, volumes 3 and 4 are missing. If anyone has access to volumes 3 and 4 please consider posting them.

This textbook directly continues the first volume of a course of geometry (M. M. Postnikov. Lectures in Geometry: Semester 1. Analytic Geometry. Moscow, Mir Publishers, 1981) based on lectures read by the author at Moscow University for students specializing in mathematics. lt contains 27 lectures, each a nearly exact reproduction of an original lecture. It treats linear algebra, with elementary differential geometry of curves and surfaces in three-dimensional space added to pave the way for further discussions.

The book was translated from Russian by by Vladimir Shokurov and was published in 1982 by Mir.

You can get the book here.

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# Contents

Preface 5

## Lecture 1 11

Vector spaces. Subspaces. Intersection of subspaces. Linear spans. A sum of subspaces. The dimension of a subspace. The dimension of a sum of subspaces. The dimension of a linear span

## Lecture 2 19

Matrix rank theorem. The rank of a matrix product. The Kronecker-Capelli theorem. Solution of systems of linear equations

## Lecture 3 28

Direct sums of subspaces. Decomposition of a space as a direct sum of subspaces. Factor spaces. Homomorphisms of vector spaces. Direct sums of spaces

## Lecture 4 36

The conjugate space. Dual spaces. A second conjugate space. The transformation of a conjugate basis and of the coordinates of covectors. Annulets. The space of solutions of a system of homogeneous linear equations

## Lecture 5 47

An annulet of an annulet and annulets of direct summands. Bilinear functionals and bilinear forms. Bilinear functionals in a conjugate space. Mixed bilinear functionals. Tensors

## Lecture 6 58

Multiplication of tensors. The basis of a space of tensors. Contraction of tensors. The rank space of a multilinear functionals

## Lecture 7 64

The rank of a multilinear functional. Functionals and permutations. Alternation

## Lecture 8 72

Skew-symmetric multilinear functionals. External multiplication. Grassman algebra. External sums of covectors. Expansion of skew-symmetric functionals with respect to the external products of covectors of a basis

## Lecture 9 82

The basis of a space of skew-symmetric functionals. Formulas for the transformation of the basis of that space. Multivectors. The external rank of a skew-symmetric functional. Multivector rank theorem. Conditions for the equality of multivectors

## Lecture 10 92

Cartan’s divisibility theorem. Plücker relations. The Plücker coordinates of subspaces. Planes in an affine space. Planes in a projective space and their coordinates

## Lecture 11 106

Symmetric and skew-symmetric bilinear functionals. A matrix of symmetric bilinear functionals. The rank of a bilinear functional. Quadratic functionals and quadratic forms. Lagrange theorem

## Lecture 12 118

Jacobi theorem. Quadratic forms over the fields of complex and real numbers. The law of inertia. Positively definite quadratic functionals and forms

## Lecture 13 127

Second degree hypersurfaces of an n-dimensional projective space. Second degree hypersurfaces in a complex and a real-complex projective space. Second degree hypersurfaces of an n-dimensional affine space. Second degree hypersurfaces in a complex and a real-complex affine space

## Lecture 14 140

The algebra of linear operators. Operators and mixed bilinear functionals. Linear operators and matrices. Invertible operators. The adjoint operator. The Fredholm alternative. Invariant subspaces and induced operators

## Lecture 15 151

Eigenvalues. Characteristic roots. Diagonalizable operators. Operators with simple spectrum. The existence of a basis in which the matrix of an operator is triangular. Nilpotent operators

## Lecture 16 160

Decomposition of a nilpotent operator as a direct sum of cyclic operators. Root subspaces. Normal Jordan form. The Hamilton-Cayley theorem

## Lecture 17 170

Complexification of a linear operator. Proper subspaces belonging te characteristic roots. Operators whose complexification is diagonalizable

## Lecture 18 179

Euclidean and unitary spaces. Orthogonal complements. The identification of vectors and covectors. Annulets and orthogonal complements. Bilinear functionals and linear operators. Elimination of arbitrariness in the identification of tensors of different types. The metric tensor. Lowering and raising of indices

## Lecture 19 191

Adjoint operators. Self-adjoint operators. Skew-symmetric and skew-Hermitian operators. Analogy between Hermitian operators and real numbers. Spectral properties of self-adjoint operators. The orthogonal diagonalizability of self-adjoint operators

## Lecture 20 199

Bringing quadratic forms into canonical form by orthogonal transformation of variables. Second degree hypersurfaces in a Euclidean point space. The minimax property of eigenvalues of self-adjoint operators. Orthogonally diagonalizable operators

## Lecture 21 208

Positive operators. Isometric operators. Unitary matrices. Polar factorization of invertible operators. A geometrical interpretation of polar factorization. Parallel translations and centroaffine transformations. Bringing a unitary operator into diagonal form. А rotation of an n-dimensional Euclidean space as a composition of rotations in two-dimensional planes

## Lecture 22 221

Smooth functions. Smooth hypersurfaces. Gradient. Derivatives with respect to a vector. Vector fields. Singular points of a vector field. A module of vector fields. Potential and irrotational vector fields. The rotation of a vector field. The divergence of a vector field. Vector analysis. Hamilton’s symbolic vector. Formulas for products. Compositions of operators

## Lecture 23 243

Continuous, smooth, and regular curves. Equivalent curves. Regular curves in the plane and graphs of functions. The tangential hyperplane of a hypersurface. The length of a curve. Curves in the plane. Curves in a three-dimensional space

## Lecture 24 262

Projections of a curve onto the coordinate planes of the moving n-hedron. Frenet’s formulas for a curve in the n-dimensional space. Representation of a curve by its curvatures. Regular surfaces. Examples of surfaces

## Lecture 25 276

Vectors tangential to a surface. The tangential plane. The first
quadratic form of a surface. Mensuration of lengths and angles on a surface. Diffeomorphisms of surfaces. Isometries and the intrinsic geometry of a surface. Examples. Developables

## Lecture 26 291

The tangential plane and the normal vector. The curvature of a normal section. The second quadratic form of a surface. The indicatrix of Dupin. Principal curvatures. The second quadratic form of a graph. Ruled surfaces of zero curvature. Surfaces of revolution

## Lecture 27 310

Weingarten’s derivation formulas. Coefficients of connection. The Gauss theorem. The necessary and sufficient conditions of isometry

Subject Index 346