## Lecture Notes

### Differential Geometry

• #### Problem Sets

• Click on the following links for problems.

• #### Classical Differential Geometry

• Modern Differential Geometry
• #### References and Further Reading

• A. N. Aliev, J. Kalayci, and Y. Nutku, General minimal surface solution for gravitational instantons, Phys. Rev. D 56, 1332–1333 (1997)
Christopher Anand, Ward's solitons
Christopher Kumar Anand, Uniton Bundles
P. L. Antonelli, P. S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Springer, 1993
P. L. Antonelli, R. Miron (eds.), Lagrange and Finsler Geometry, Applications to Physics and Biology, Springer, 1996
G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories, D. Reidel Publishing Company, 1985
Michael Atiyah, Paul Sutcliffe, Skyrmions, instantons, mass and curvature
John Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, 1994
Артур Бессе, Многообразия Эйнштейна, Мир, 1990

This is the Russian translation of

Arthur L. Besse, Einstein Manifolds, Springer-Verlag

published by Mir Publishers, Moscow. Arthur Besse is not a person but a group of French differential geometers like Nicolas Bourbaki being a group of French mathematicians. As far as I know, the original book was published in English as cited above. I don't have a copy of the original book but its Russian translation.

William M. Boothby, Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, v. 120, 2nd Edition, Academic Press, 2002
Nicole Berline, Ezra Getzler and Michèle Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992
Roger Bielawski, Monopoles and the Gibbons-Manton metric
Jean-Luc Brylinski and Philip Foth, Complex Manifolds, Vector Bundles and Hodge Theory, Birkhauser, 1998
Benoit Charbonneau, Jacques Hurtubise, The Nahm transform for calorons
A. Comtet, Instantons and minimal surfaces, Phys. Rev. D 18, 3890–3892 (1978)
Ana Cannas da Silva, Introduction to Symplectic and Hamiltonian Geometry, Notes for a Short Course at IMPA, Rio de Janeiro, February 2002
Ana Cannas da Silva and Alan Weinstein, Geometric Models for Noncommutative Algebras, 1998
Manfredo Perdigão do Carmo, Riemannian Geometry, Birkhäuser, 1992
Gerald V. Dunne, Aspects of Chern_Simons Theory
Gerald V. Dunne, Vishesh Khemani, Numerical Investigation of Monopole Chains
A.P. Fordy and J.C. Wood (eds.), Harmonic Maps and Integrable Systems, Aspects of Mathematics, Vol. E23, Vieweg, 1994
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, Springer-Verlag, 2004
Nigel Hitchin, Monopoles, Minimal Surfaces and Algebraic Curves, Les Presses de l'Université de Montréal, 1987
Nigel Hitchin, The Wess-Zumino term for a harmonic map
N.J. Hitchin, N.S. Manton, M.K. Murray, Symmetric Monopoles
N. J. Hitchin, G. B. Segal and R. S. Ward, Integrable Systems, Twistors, Loop Groups, and Riemann Surfaces, Clarendon Press, 1999
Gerard 't Hooft, Falk Bruckmann, Monopoles, Instantons and Confinement
Conor Houghton, Nicholas Manton, Paul Sutcliffe, Rational Maps, Monopoles and Skyrmions
Theodora Ioannidou, Paul Sutcliffe, Monopoles and Harmonic Maps
Theodora Ioannidou, Paul Sutcliffe, Monopoles from Rational Maps
Theodora Ioannidou, Paul Sutcliffe, Non-Bogomolny $$\mathrm{SU}(N)$$ BPS Monopoles
Shoshichi Kobayahsi, Differential Geometry of Complex Vector Bundles, Kanô Memorial Lecture 5, Iwanami Shoten, Publishers and Princeton University Press, 1987
V. Kotacha, R. S. Ward, Integrable Yang-Mills-Higgs Equations in 3-Dimensional De Sitter Space-Time
L. J. Mason, Global anti-self-dual Yang-Mills fields in split signature and their scattering
Matilde Marcolli, Seiberg-Witten Gauge Theory
Matilde Marcolli, Seiberg-Witten Gauge Theory, Texts and Readings in Mathematics, v. 17, Hindustan Book Agency, 1999
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer-Verlag, 1991
Michael K. Murray, Monopoles
Michael K. Murray, Paul Norbury, Michael A. Singer, Hyperbolic monopoles and holomorphic spheres
Michael K. Murray, Michael A. Singer, A note on monopole moduli spaces
Gregory L. Naber, Topology, Geometry, and Gauge Fields: Foundations, Texts in Applied Mathematics, v. 25, Springer, 1997
Mikio Nakahara, Geometry, Topology and Physics, 2nd Edition, Graduate Student Series in Physics, Taylor & Francis, 2003
Charles Nash and Siddhartha Sen, Topology and Geometry for Physicists, Academic Press, Inc., 1983
Liviu I. Nicolaescu, Notes on Seiberg-Witten Theory, Graduate Studies in Mathematics Volume 28, American Mathematical Society
Barrett O'Neill, Elementary Differential Geometry, Academic Press, 1966
Barrett O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983
Paul Norbury, Magnetic monopoles on manifolds with boundary
Y. Nutku, Gravitational Instantons and Minimal Surfaces, Phys. Rev. Lett. 77, 4702–4703 (1996)
M. K. Prasad and Charles M. Sommerfield, Exact Classical Solution for the 't Hooft Monopole and the Julia-Zee Dyon, Phys. Rev. Lett. 35, 760–762 (1975)
Andrew Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer, 2002
Robert Osserman, A Survey of Minimal Surfaces, Dover Publications, Inc., 1986
C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, 1982
Dietmar Salamon, Spin Geometry and Seiberg-Witten Invariants, 1996
Patrick Shanahan, The Atiyah-Singer Index Theorem, An Introduction, Lecture Notes in Mathematics 638, Springer-Verlag, 1978
Isidore M. Singer and John A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Undergraduate Texts in Mathematics, Springer, 1976
Paul Sutcliffe, BPS Monopoles
George Svetlichny, Preparation for Gauge Theory
Jószef Szilasi, Rezso L. Lovas and Dávid Cs Kertész, Connections, Sprays and Finsler Structures, World Scientific, 2014
Peter Szekeres, A Course in Modern Mathematical Physics, Groups, Hilbert Space and Differential Geometry, Cambridge University Press, 2004
Bayram Tekin, Multi-instantons in $$\mathbb{R}^4$$ and Minimal Surfaces in $$\mathbb{R}^{2,1}$$
John A. Thorpe, Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, Springer, 1994
David Tong, TASI Lectures on Solitons
R. S. Ward,
Two Integrable Systems Related to Hyperbolic Monopoles
Tilla Weinstein, An Introduction to Lorentz Surfaces, Expositions in Mathematics 22, Walter de Gruyter, 1996
Edward Witten, Some Exact Multipseudoparticle Solutions of Classical Yang-Mills Theory, Phys. Rev. Lett. 38, 121–124 (1977)