GEOMETRY, GROUPS, AND CONTROL
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Here is a short list of essential reading:
LEVEL one
(beginners)
1.1 General
H. Weyl,
Symmetry
, Princeton Univ Press, 1950 (renewed 1980).
I. Stewart,
Why Beauty Is True. A History of Symmetry
, Basic Books, 2007.
I.M. Yaglom,
Felix Klein and Sophus Lie. Evolution of the Idea of Symmetry in the Nineteenth Century
, Birkhauser, 1988.
J. Stillwell,
Mathematics and Its History
(3/E), Springer, 2010.
1.2 Groups (elementary)
M.A. Armstrong,
Groups and Symmetry,
Springer, 1988.
R.P Burn,
Groups. A Path to Geometry,
Cambridge Univ Press, 1985 (digital printing 2001).
V. Climenhaga, A. Katok,
From Groups to Geometry and Back,
AMS, 2017.
P.M. Neumann, G.A. Stoy, E.C. Thompson,
Groups and Geometry
, Oxford Univ Press, 1994.
1.3 Geometry (elementary)
J. Stillwell,
The Four Pillars of Geometry,
Springer, 2005.
A.B. Sossinsky,
Geometries
, AMS, 2012.
E.G. Rees,
Notes on Geometry
, Springer, 1983.
J. Stillwell,
Geometry of Surfaces
, Springer, 1992.
LEVEL two
(intermediate)
2
.1 General (algebra, topology)
M.A. Armstrong,
Basic Topology
, Springer, 1983.
W.S. Massey,
Algebraic Topology: An Introduction
, Springer, 1967.
K. Erdmann, M.J. Wildon,
Introduction to Lie Algebras
, Springer, 2006.
E.B. Vinberg,
A Course in Algebra
, AMS, 2003.
2
.2 Linear Lie Groups
J. Stillwell,
Naive Lie Theory,
Springer, 2008.
K. Tapp,
Matrix Groups for Undergraduates,
2/E, AMS, 2016.
W. Rossmann,
Lie Groups. An Introduction Through Linear Groups,
Oxford Univ Press, 2002.
A. Baker,
Matrix Groups. An Introduction to Lie Group Theory
, Springer, 2002.
2
.3 Differential Geometry (classical)
A. Pressley,
Elementary Differential Geometry,
2/E,
Springer, 2010.
M. do Carmo,
Differential Geometry of Curves and Surfaces,
Prentice-Hall, 1976.
W. Kuhnel,
Differential Geometry: Curves--Surfaces--Manifolds
, 3/E, AMS, 2015.
A. Gray, E. Abbena, S. Salamon,
Modern Differential Geometry of Curves and Surfaces with MATHEMATICA
, 3/E, Chapman & Hall/CRC, 2006.
LEVEL three
(advanced)
3
.1 General (topology, geometry)
John M. Lee,
Introduction to Smooth manifolds
, Springer, 2003.
Jeffrey M. Lee,
Manifolds and Differential Geometry
, AMS, 2009.
P.J. Olver,
Equivalence, Invariants, and Symmetry
, Cambridge Univ Press, 1995.
P.W. Michor,
Topics in Differential Geometry
, AMS, 2008.
B. Rosenfeld,
Geometry of Lie Groups
, Kluwer, 1987.
3
.2 Lie Groups (and Lie Algebras)
A. Arvanitoyeorgos,
An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
, AMS, 2003.
A.L. Onishchik, E.B. Vinberg,
Foundations of Lie Theory
(in "Lie Groups and Lie Algebras I", A.L. Onishchik (ed.)), Springer, 1993.
V.V. Gorbatsevich, A.L. Onishchik,
Lie Transformation Groups
(in "Lie Groups and Lie Algebras I", A.L. Onishchik (ed.)), Springer, 1993.
J.J. Duistermaat, J.A.C. Kolk,
Lie Groups,
Springer, 2000.
J. Hilgert, J.H. Neeb,
Structure and Geometry of Lie Groups
, Springer, 2012.
3
.3 Geometry (differential-geometric structures)
R.W. Sharpe,
Differential Geometry. Cartan's Generalization of Klein's Erlangen Program,
Springer, 1997 (corr. 2nd printing, 2000) .
W. M. Goldman,
Geometric Structures on Manifolds
, AMS, 2022.
S. Helgason,
Differential Geometry, Lie Groups, and Symmetric Spaces
, AMS,1978 (corr. printings, 2001, 2012).
J. N. Clelland,
From Frenet to Cartan: The Method of Moving Frames
, AMS, 2017.
A. Bejancu, H.R. Farran,
Foliations and Geometric Structures
, Springer, 2006.
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