Introduction of Global calculus
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Global Calculus
S. Ramanan
Graduate Studies
in Mathematics
Volume 65
American Mathematical Society
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Contents
Contents 3
Preface 5
I Sheaves and Dierential Manifolds:
Denitions and Examples 9
1 Sheaves and Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Dierential Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Lie Groups; Action on a Manifold . . . . . . . . . . . . . . . . . . . . . . 30
II Dierential Operators 33
1 First Order Dierential Operators . . . . . . . . . . . . . . . . . . . . . 33
2 Locally Free Sheaves and Vector Bundles . . . . . . . . . . . . . . . . . 35
3 Flow of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Theorem of Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Tensor Fields; Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . 57
6 The Exterior Derivative; de Rham Complex . . . . . . . . . . . . . . . . 62
7 Dierential Operators of Higher Order . . . . . . . . . . . . . . . . . . . 69
Appendices 81
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4 CONTENTS
preface
This book is intended for postgraduate and ambitious senior undergraduate students.
The key word is Dierential Operators. I have attempted to develop the calculus of such
operators in an uncompromisingly global set - up.
I have tried to make the book as self - contained as possible. I do make heavy
demands on the algebraic side, but at least recall the required results in a detailed way
in an appendix.
This book has had an unusually long gestation period. It is incredible for me to
realise that the general tone of the book is very much the same as that of a course which
I gave way back in 1970 at the Tata Institute, which I (hopefully) improved upon a
couple of years later, and expanded into a two quarter graduate course at the University
of California, Los Angeles, in 1979 - 80. The encouragement that the audience gave me,
especially Nagisetty Venkateswara Rao (now at the University of Ohio, Toledo), my rst
student the late Annamalai Ramanathan, and Jost at UCLA, was quite overwhelming.
The decision to write it all up was triggered by a strong suggestion of Stefan Mueller -
Stach (now at Mainz University). He drove me from Bayreuth to Trieste in Italy, and I
used the occasion to clear some of his doubts in mathematics. Apparently happy at my
eort, he suggested that I had a knack for exposition and should write books. I took
this for more than ordinary politeness, and embarked on this project nearly ten years
later.
Chanchal Kumar, Amit Hogadi and Chaitanya Guttikar enthusiastically read por-
tions of the book and made many constructive suggestions. Madhavan of the Chennai
Mathematical Institute and Nandagopal of the Tata Institute of Fundamental Research
helped with the gures. Ms. Natalya Pluzhnikov of the American Mathematical Society
took extraordinary care in nalising the copy. All through, and particularly in the last
phase, my wife Anu’s support, physical and psychological, was invaluable.
I utilised the hospitality of various institutions during the course of writing. Besides
the Tata Institute of Fundamental Research, my Alma Mater, I would like to mention
specially the Institute of Mathematical Sciences at Chennai, the International Centre for
Theoretical Physics, Trieste, and at the nal stage, Consejo Superior de Investigaciones
Cientícas in Madrid.
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6 CONTENTS
I wish to thank all the individuals and institutions for the help received.
I collaborated for long years with M. S. Narasimhan. I wish to take this opportunity
to acknowledge the exciting time that I spent doing mathematical research with him.
Specically, the idea of what I call the composition formula in the last chapter of this
book, arose in our discussions. Narasimhan also suggested appropriate references for
Chapter 8.
I learnt modern Dierential Geometry from Jean - Louis Koszul. His lectures at the
Tata Institute, of which I took notes, were an epitome of clarity. I wish to thank him
for choosing to spend time in his youth to educate students in what must have seemed
at that time the outback.
There is a short summary at the start of each chapter explaining the contents. Here
I would like to draw attention to what I think are the new features in my treatment.
In Chapter 1 sheaves make their appearance before dierential manifolds. Everyone
knows that this is the ‘correct’ denition but I know of few books that have adopted this
point of view. The reason is that generally sheaves are somehow perceived to be more
dicult to swallow at the outset. I believe otherwise. In my experience, if sucient
motivation is provided and many illustrative examples given, students take concepts in
their stride, and will in fact be all the better equipped in their mathematical life for an
early start.
In Chapter 2 I have introduced the notion of the Connection Algebra and believe
most computations can, and ought to be, made in this algebra.
In Chapter 3 the treatment of densities and orientation are, I believe, nonconven-
tional. In particular, the change - of - variable formula is not used but is in fact a simple
consequence of this approach.
Chapter 4 is a fairly straightforward account of sheaf cohomology.
Connections are treated as tools to lift symbols to dierential operators; various
tensors connected with connections and linear connections have a natural interpretation
from this point of view. These are worked out in Chapter 5. The existence of torsion
free linear connections compatible with various structures lead to natural integrability
conditions on them. This treatment, which I believe is new, is given in Chapter 6.
The additional structures are themselves studied in Chapter 7 with some emphasis
on naturally occurring dierential operators.
Chapter 8 is an account of the local theory of elliptic operators, while Chapter 9
contains the composition formula I mentioned earlier. A general vanishing theorem
for harmonic sections of an elliptic complex is proved here under a suitable curvature
hypothesis. The Bochner, Lichnerowicz and Kodaira vanishing theorems are derived as
special cases, followed by a short account of how Kodaira’s vanishing theorem leads to
the imbedding theorem.
Mistakes, particularly relating to signs and constants, are a professional hazard. I
Chapter : CONTENTS
would be grateful to any reader who takes trouble to inform me of such and other errors,
misleading remarks, obscurities, etc.
October 2004 Sundararaman Ramanan
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