## Lecture Notes

### Differential Geometry

#### Lecture Notes

#### Classical Differential Geometry

#### Modern Differential Geometry

#### Einstein Manifolds

#### Problem Sets

#### Classical Differential Geometry

#### Modern Differential Geometry

#### References and Further Reading

Frame Fields

Connection Forms

Structural Equations

The Curvature of a Curve in Euclidean 3-Space \(\mathbb{R}^3\)

The Curvature of a Surface in Euclidean 3-Space \(\mathbb{R}^3\)

The interior area of a simple closed curve is invariant under a rigid motion

Matrix Lie Groups

The Lie Algebra of the Orthogonal Group \(\mathrm{O}(n)\) (\(\mathrm{SO}(n)\))

Orthogonal Group \(\mathrm{O}(n)\) and Symmetry

The Lorentz Group

Lie Brackets (for \(n\times n\) Matrices)

More Examples of Lie Groups: 3-Sphere as a Lie group, The 3-Dimensional Heisenberg Group

Lie Group Actions and Lie Group Representations

Irreducible Representations of \(\mathrm{U}(1)\)

The Representations of \(\mathrm{SU}(2)\)

Differentiable Manifolds and Tangent Spaces

The Diagonalization of a Riemannian Metric

Tensors I

Fibre Bundles

Vector Bundles

Line Bundles

Sections of a Line Bundle I

Sections of a Line Bundle II: Gauge Potential, Gauge Transformation, and Field Strength

Parallel Transport, Holonomy, and Curvature

Click on the following links for problems.

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 the book by Arthur L. Besse, *Einstein Manifolds*. (The original book was published in English by Springer-Verlag.) Arthur Besse is not a person but the psuedonym of a group of French differential geometers. The reason I cite the Russian translation is that that is what I am actually using. I don't have a copy of the original book but only its Russian translation.

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)