Transformation groups in differential geometry. Reprint of the 1972 ed. (English) Zbl 0829.53023
Classics in Mathematics. Berlin: Springer-Verlag. viii, 182 p. (1995).
[For a review of the original edition, see Zbl 0246.53031.]
Since its original publication in 1972 as part of the series “Ergebnisse der Mathematik und ihrer Grenzgebiete”, this book has become an important reference work on fundamental results about transformation groups and related problems in differential geometry. This softcover edition appears in the series “Classics in mathematics” which is devoted to reprinting important monographs in comparedly inexpensive editions.
The book consists of four chapters: I. Automorphisms of \(G\)-structures; II. Isometries of Riemannian manifolds; III. Automorphisms of complex manifolds; IV. Affine, conformal and projective transformations.
Chapter I is primarily concerned with the question of when the automorphism group of a \(G\)-structure admits a Lie group structure. Moreover, the concept of pseudogroup structure is introduced and the relationship between the two concepts is investigated. In Chapter II it is shown that the isometry group of a Riemannian manifold is a Lie group. Riemannian manifolds with large groups of isometries as well as with little isometries are examined and classified. Other topics covered in this chapter are fixed points of isometries and the connection between infinitesimal isometries and characteristic numbers. Chapter III covers complex manifolds, in particular compact complex manifolds with finite transformation groups, compact Einstein-Kähler-manifolds, compact Kähler manifolds with constant scalar curvature resp. with nonpositive first Chern class, and zeros of holomorphic vector fields. Chapter IV is concerned with automorphisms of affine, projective and conformal connections.
The book contains an extensive bibliography which also covers subjects not treated in the book (such as homogeneous spaces). For bibliographies that also cover the time after 1972, see [A. V. Aminova, J. Sov. Math. 55, No. 5, 1996-2041 (1991; Zbl 0735.53026)] or [B. A. Dubrovin, S. P. Novikov and A. T. Fomenko, Modern geometry. Methods and applications, 2nd rev. ed., Moskva (1986; Zbl 0601.53001)].
Since its original publication in 1972 as part of the series “Ergebnisse der Mathematik und ihrer Grenzgebiete”, this book has become an important reference work on fundamental results about transformation groups and related problems in differential geometry. This softcover edition appears in the series “Classics in mathematics” which is devoted to reprinting important monographs in comparedly inexpensive editions.
The book consists of four chapters: I. Automorphisms of \(G\)-structures; II. Isometries of Riemannian manifolds; III. Automorphisms of complex manifolds; IV. Affine, conformal and projective transformations.
Chapter I is primarily concerned with the question of when the automorphism group of a \(G\)-structure admits a Lie group structure. Moreover, the concept of pseudogroup structure is introduced and the relationship between the two concepts is investigated. In Chapter II it is shown that the isometry group of a Riemannian manifold is a Lie group. Riemannian manifolds with large groups of isometries as well as with little isometries are examined and classified. Other topics covered in this chapter are fixed points of isometries and the connection between infinitesimal isometries and characteristic numbers. Chapter III covers complex manifolds, in particular compact complex manifolds with finite transformation groups, compact Einstein-Kähler-manifolds, compact Kähler manifolds with constant scalar curvature resp. with nonpositive first Chern class, and zeros of holomorphic vector fields. Chapter IV is concerned with automorphisms of affine, projective and conformal connections.
The book contains an extensive bibliography which also covers subjects not treated in the book (such as homogeneous spaces). For bibliographies that also cover the time after 1972, see [A. V. Aminova, J. Sov. Math. 55, No. 5, 1996-2041 (1991; Zbl 0735.53026)] or [B. A. Dubrovin, S. P. Novikov and A. T. Fomenko, Modern geometry. Methods and applications, 2nd rev. ed., Moskva (1986; Zbl 0601.53001)].
Reviewer: A.Perović (Berlin)
MSC:
| 53C10 | \(G\)-structures |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
