Differential geometry and Lie groups
for Physicists - what is where in video-lectures
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Lecture 1 Dual space, dual basis, transformation of bases and components

0:00:00  Organizational information
0:03:25  What is Mathematical Physics (and which part will be here)
0:07:50  Raoul Bott on linear algebra (quotation)
0:16:40  Change-of-basis matrix
0:27:40  Dual space L^*
0:35:55  Dual basis (in L^*)
1:00:40  Bilinear (canonical) pairing
1:08:45  The dual basis for a new basis in L
1:17:50  Transformation of components of vectors and co-vectors
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Lecture 2 Tensors as maps, tensor product, decomposition w.r.t. basis

0:02:30  The second dual of L (canonically isomorphic to L)
0:18:05  Multilinear map
0:26:00  Tensors as multilinear maps
0:29:55  Components of a tensor
0:34:15  Transformation of components of a tensor
0:48:30  Several roles of the same tensor
0:51:45  A short note on fundamentalism
0:53:15  Tensor operations: Linear combination and contraction
1:13:55  Tensor operations: Tensor product
1:22:20  A basis of tensors, decomposition of a tensor w.r.t. the basis
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Lecture 3 Metric and cometric, structured set, standard matrix groups

0:00:40  Metric tensor
0:08:45  The cometric
0:21:10  Lowering and raising of indices
0:37:30  Definition of a group
0:40:30  Group of bijections of a set
0:43:55  Bijections of structured sets
0:50:00  Linear structure and the group Aut V = GL(V)
0:57:20  The group GL(n,R) and its isomorphisms with GL(V)
1:07:50  Adding a bilinear form
1:17:40  Symmetric non-degenerate case, groups O(r,s) and O(n)
1:25:20  Volume in V and SL(V), SL(n,R), SO(r,s) and SO(n)
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Lecture 4 Affine, Euclidean and Poincaré groups as matrix groups

0:08:30  Hermitian scalar product and the groups U(n) and SU(n)
0:18:00  Affine group
0:24:40  Euclidean and Poincare groups
0:30:20  Affine (Euclidean, Poincare) group as matrix groups
0:40:00  Direct product G_1 x G_2 of groups
0:55:00  GL_+(n,R) = SL(n,R) x GL(1,R)
1:11:20  Rough idea of a manifold


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Lecture 5 Manifolds and smooth maps, atlas for S^n and RP^n

0:00:30  Manifold - chart (local coordinates), smooth atlas
0:10:00  Angles as coordinates on spheres
0:20:40  Stereographic projection - other atlas (coordinates) on spheres
0:34:00  Projective space RP^n - definition and atlas
0:48:05  CP^n, atlas, complex manifold
0:52:50  Cartesian product of manifolds
0:58:25  Smooth maps of manifolds
1:09:35  Diffeomorphism, embedding, submanifold


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Lecture 6 Surfaces, curve and function, Lie group

0:00:15  Immersion, embedding, submanifold (again, a bit more)
0:08:00  Whitney theorem
0:13:00  Surfaces in Cartesian spaces (constraints, parametrization)
0:33:00  Hypersurfaces
0:38:45  SL(n,R) as a hypersurface
0:44:20  A curve and a function on a manifold
0:56:35  The concept of a Lie group
1:17:55  Lie algebras (first steps, no connection with groups)
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Lecture 7 Lie algebra (general), the Lie algebra of a (matrix) Lie group

0:01:10  Lie algebras - still with no connection to groups
0:05:45  Lie algebras from associative algebras
0:20:15  Poisson bracket as a commutator in a Lie algebra
0:27:30  Structure constants
0:46:15  The Lie algebra of a Lie group
1:05:10  Standard notations
1:07:10  Computation of Lie algebras gl(n,R), o(n), u(n), sl(n,R)

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Lecture 8 1-parameter subgroups (and examples), exponential mapping

0:00:05  1-parameter subgroups
0:08:35  The differential equation for the 1-parameter subgroup
0:12:30  Solution of the equation (how any 1-parameter subgroup looks like)
0:23:20  1-parameter subgroups on SO(2)
0:33:45  Exponential mapping, exponential Lie groups
0:38:15  Exponential mapping for O(2)
0:49:30  1-parameter subgroups on SU(2)
1:12:10  1-parameter subgroups on SO(3)
1:24:30  1-parameter subgroup as a geodesic

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Lecture 9 Derived homomorphism, representation of a group and Lie algebra

0:00:05  mapping of 1-parameter subgroup w.r.t. a homomorphism
0:07:20  The corresponding commutative diagram for f and derived f'
0:11:40  Algorithm for computing f' from (known) f
0:17:20  f' is a Lie algebra homomorphism
0:23:15  Example: f = det, f' = Tr
0:36:40  Representations of groups - basics
0:41:30  Representation rho (A) = A (for matrix groups)
0:44:50  Examples: rho(A) =1, rho(A) = det A, rho(A)B=ABA^{-1}
0:53:00  Derived representation rho' (of the Lie algebra)
1:02:35  Example: For rho(A)B=ABA^{-1} we get rho'(C)B=CB-BC
1:11:35  Conjugation in a group G and its properties
1:18:50  The origin of Ad and ad representations

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Lecture 10 Irreducible representation, intertwining operator, Schur's lemma

0:00:05  Recapitulation: conjugation, representations, Ad and ad
0:09:15  Explicit formulas how Ad and ad representations operate
0:18:35  Some useful notations for matrices of representations
0:23:10  Generator of a representation, in particular for Ad
0:31:10  Isomorphism of Lie algebras (su(2) and so(3) are such)
0:40:30  Reducible and irreducible representations
0:53:00  Example: rho(A)=A is irreducible for SO(2)
1:05:55  Intertwining operator (equivariant mapping)
1:13:10  Equivalent representations and (part of) Schur's lemmas

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Lecture 11 Killing-Cartan form, tensor product of representations

0:00:01  Correction of a small mistake in Lecture 10
0:02:10  The Schur's lemma finished
0:16:10  What do all irreducible representations of U(1) look like
0:27:25  Invariant scalar product
0:45:50  Killing-Cartan form, its Ad-invariance, semi-simple Lie algebra
1:08:00  Tensor product of vector spaces
1:18:35  Tensor product of operators
1:23:10  Tensor product of representations of a group
1:26:45  Its derived representation (of the Lie algebra)

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Lecture 12 Vector at P on M, vector field

0:04:15  Curves on M tangent at P
0:12:20  Tangency at P as equivalence (independent of coordinates)
0:18:50  Vectors at point P as equivalence classes, linear combination
0:27:00  Dimension of T_PM (tangent space at P), its "coordinate" basis
0:37:20  Directional derivative, vector as a linear functional on F(M)
0:58:35  Change of basis and components under change of coordinates
1:04:40  Other two definitions of vector (including the "classical" one)
1:10:50  Vector field, (coordinate) basis and components
1:18:33  Vector field as a linear operator on F(M)
1:27:20  Example in the plane R^2[x,y]

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Lecture 13 Integral curves, flow, tensor fields, gradient

0:04:30  Integral curves of a vector field
0:13:00  System of n ordinary quasi-linear autonomous equations for x(t)
0:15:50  Example 1: "Rotation" vector field in plane
0:30:00  Example 2: "Spiraling" on a cylinder
0:31:15  Example 3: Meridians and parallels on a sphere as integral curves
0:33:00  Flow of a vector field
0:41:40  Tensor fields on a manifold
0:47:35  Vector fields as a module over algebra F(M)
1:05:55  R-linearity versus F(M)-linearity
1:12:00  Gradient df of a function f (as a covector field)
1:17:00  Gradients of coordinate functions as a basis for covectors
1:27:35  Coordinate basis for arbitrary tensor fields
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Lecture 14 Metric tensor, length of a curve, orthonormal (co)frame

0:03:20  Canonical tensor fields on M
0:05:25  Metric tensor on M, Riemannian manifold (M,g)
0:07:50  Euclidean manifold En (space), Minkowski manifold
0:19:25  Length of a curve in E3
0:26:00  Length of a curve on general (M,g)
0:31:30  Frame and coframe fields on M (repere mobile)
0:43:00  Orthonormal frame and coframe on (M,g)
0:55:10  Metric tensor in orthonormal frame
1:01:25  Orthogonal coordinates
1:06:20  Geodesics as the shortest paths
1:12:00  Kinetic energy and metric tensor
1:23:25  Gradient of a function f as a vector field

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Lecture 15 Pull-back and push forward operations

0:02:30  Pull-back of functions (00-type tensor fields)
0:18:00  Push-forward of vectors (individual 10-type tensors)
0:30:05  Problems with push-forward of a vector field
0:35:15  Pull-back of covector fields (01-type tensor fields)
0:47:00  On education of differential geometry in England
0:48:00  Push-forward of vectors - another definition
0:52:00  Pull-back of 0p-type tensor fields
1:02:25  Summary of what has already been done
1:06:50  Pull-back and push-forward for diffeomorphisms
1:17:35  Their important properties

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Lecture 16 Induced metric, Lie transport, Lie derivative

0:02:00  Length of a vector versus length of the corresponding curve
0:09:45  Induced metric tensor
0:18:30  Example: g on the unit sphere from standard embedding
0:26:35  General coordinate expression
0:32:00  Curved and flat toruses
0:40:20  Rotational surface (e.g. cone)
0:42:10  Pseudosphere (space-like surface)
0:47:50  Kinetic energy in Lagrangian mechanics
0:57:20  Potential energy in Lagrangian mechanics
1:02:25  Flow of a vector field again (see Lecture 13)
1:05:00  Its composition property
1:11:55  Lie transport as pull-back w.r.t. flow
1:21:25  Lie derivative introduced (see also 1:28:05)
1:22:15  Rotational and translation invariance as Lie invariances

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Lecture 17 Properties and computation of Lie derivative

0:01:45  Lie derivative of linear combination and tensor product
0:06:30  Lie derivative of a general tensor field - beginning
0:15:40  Lie derivative of a function
0:19:40  Lie derivative of a coordinate (vector and covector) basis
0:28:20  Lie derivative of a 11-tensor field (general pattern visible)
0:37:50  Lie derivative of a vector field and commutator (Lie bracket)
0:48:25  Commutators in quantum mechanics
0:51:50  Exponent of Lie derivative (and pull-back of a flow)
1:00:30  Straightening out lemma
1:04:50  A proof of the exponent of Lie derivative
1:14:25  Non-coordinate (non-holonomic) frame fields and commutator
1:22:00  Example: Orthonormal polar frame is non-holonomic

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Lecture 18 Tensor algebra, interpretation of commutator of vector fields

0:00:00  Tensor algebra
0:06:00  Direct sum of vector spaces
0:11:30  Summation of apples and pears finally allowed
0:14:20  Product in tensor algebra
0:17:40  Automorphisms and derivations of tensor algebra
0:26:30  ZxZ-grading of the tensor algebra
0:44:10  Commutator of derivations, Lie algebra of derivations
0:52:00  Geometric interpretation of commutator of vector fields
1:14:15  Example: Orthonormal polar frame in the plane
1:22:35  Example: Orthonormal frame on the 2-sphere
1:26:30  Example: Rotations in 3D

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Lecture 19 Isometries of Riemannian manifolds and Killing equations

0:00:35  Diffeomorphisms which preserve length of curves
0:13:30  Isometries of a Riemannian manifold
0:17:30  Derivation of Killing equations
0:20:15  Reflections are out of scope of this method
0:22:25  Killing vectors form a Lie algebra (of the group of isometries)
0:26:30  Coordinate expression of Killing equations
0:38:30  Example: Killing vectors for 2D-Euclidean plane
0:59:15  Commutator of two Killings - how to use it to find more solutions
1:02:00  Example: Killing vectors for 2D-flat torus (local Lie algebra)
1:10:10  Example: Killing vectors for 2D-rotational surfaces
1:13:25  Example: Killing vectors for 2D-curved torus
1:16:25  Example: Killing vectors for 2D-round sphere

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Lecture 20 Killings for pseudo-Euclidean space, conformal Killings

0:00:05  Example: Killing vectors for pseudo-Euclidean space
0:34:05  What flows they produce
0:39:20  Hyperbolic rotations
0:47:30  Minkowski spacetime: hyperbolic rotations are boosts
0:53:10  Conformal transformations of (M,g)
1:05:30  Conformal Killing equations and vectors
1:11:20  Example: Dilation field as conformal Killing
1:16:55  Example: 2D Euclidean plane and Cauchy-Riemann relations
1:27:10  Tensor fields with prescribed symmetry

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Lecture 21 Action of a group on a manifold, orbits, stabilizers

0:02:55  Right and left action of G on M
0:10:40  Example: Actions of GL(n,R) on Rn
0:15:40  Example: Actions of G on G (left and right translations, conjugation)
0:21:50  General trick: Right action from a left on (and vice versa)
0:28:15  Orbit Om given by a point m on M
0:36:00  Example: Orbits for actions of GL(n,R) and SO(n) on Rn
0:55:50  Stabilizer Gm of a point m on M
1:03:20  Stabilizers Gm and Gm' if m,m' are on common orbit
1:17:10  Example: Stabilizer of the north pole of a sphere

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Lecture 22 Fundamental fields, generators of an action of G on M

0:02:40  Representation as an action (specific features)
0:06:30  Representation of G (in F(M)) from an action of G on M
0:15:40  The same representation as a pull-back
0:26:40  The flow on M given by a 1-parameter subgroup exp(tX)
0:29:00  Fundamental field, generators of the action
0:34:35  The derived representation (in terms of fundamental fields)
0:42:00  Properties of fundamental fields (and generators)
0:48:00  Example: All that for the action of GA(1,R) on R
1:11:30  Example: Generators of the action of SO(3) on R3

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Lecture 23 Regular representation, left cosets, G/H

0:00:10  Example: Generators of rotations and translations on R3
0:07:00  Relation to operators of momentum and angular momentum in QM
0:11:20  The regular representation of G (on functions on G)
0:28:45  Spherical harmonics regarded as functions on SO(3)
0:32:15  G-space, transitive action, homogeneous space, free action
0:41:15  Example: Right action of GL(n,R) on frames in L
0:52:15  Orbits of free action
0:59:00  Orbits of non-free action, left cosets
1:07:40  Useful picture to visualize left cosets, fibre bundle
1:21:00  G/H as a homogeneous space for G

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Lecture 24 M as G/H, factor-group, homomorphism theorem

0:00:10  Points of a homogeneous space and left cosets (bijection)
0:06:30  Natural mapping from M to G/H
0:16:10  Any G-homogeneous space M is isomorphic to G/H for some H
0:22:00  Example: Spheres S^n = SO(n)/SO(n-1)
0:28:20  Example: Spheres S^{2n-1} = SU(n)/SU(n-1)
0:35:50  Principal homogeneous space
0:45:25  G/H as a group (factor-group), normal subgroup
0:52:20  Example: SO(2) in SO(3) is not normal
1:01:00  Homomorphism theorem (for groups)
1:12:30  Example: GL(n)/SL(n) = GL(1)
1:17:30  Example: Z/3Z = Z_3
1:22:15  Discrete kernel case, covering (beginning)

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Lecture 25 SU(2) as double cover of SO(3), some consequences

0:00:10  Discrete kernel case, covering (continuation)
0:06:30  Example: R as (infinite) cover of U(1)
0:19:15  2-sheeted covering homomorphism of SU(2) to SO(3)
0:39:10  Useful properties of Pauli matrices
0:44:15  Computing R(A) (rotation R given by A from SU(2))
0:57:00  Explicit formula for R(A)
0:59:55  Computing the kernel of the homomorphism, the covering
1:15:00  1-st consequence: Lie algebras are isomorphic
1:18:55  2-nd consequence: Lifted representations
1:25:30  Representation rho (A) = A of SU(2) is not lifted from SO(3)
1:30:20  3-rd consequence: Non-contractible loops on SO(3)