Название спецкурса на русском языке
Конформная геометрия и римановы поверхности
Перевод названия курса на английский язык
Conformal Geometry and Riemann Surfaces
Целевая аудитория
4 курс
5 курс
6 курс
Магистранты
Аспиранты
Подразделение
[Кафедра высшей геометрии и топологии]
Семестр
Полгода (весна)
Тип курса
Спецкурс на английском языке
Учебный год
2023/24
День недели
понедельник
Время
16:45-18:20
Формат проведения
В аудитории
Аудитория
[Ещё не назначена]
Аннотация
1) Conformal transformations. Definition. Infinitesimal conformal
transformations. Lie derivative of metric tensor with respect to a
generator of conformal transformation.
2) Mobius transformations of R^n. The Lie algebra of Mobius group.
Theorem: For n>=3 the Lie algebra of local conformal vector fields in R^n
is finite-dimensional and coincide with the Lie algebra of Mobius group.
3) Mobius transformations of R^n from isometries of R^{n+1,1}.
4) Weil, Schouten and Cotton tensors. The formula expressing the Riemann
tensor in terms of metric and Ricci tensors for n=3. Transformations
properties of the Weil and Cotton tensors under conformal changes of
Riemannian metric.
5) Necessary and sufficient conditions for conformal flatness (without
proof).
6) Isothermal coordinate on two-dimensional surfaces. Beltrami equation.
All 2-dimensional Riemannian manifolds are conformally flat. Local
conformal maps are holomorphic or anti-holomorphic maps.
7) Genus of hyperelliptic Riemann surfaces. Vector fields and
differentials on Riemann surfaces. Holomorphic differentials on
hyperelliptic Riemann surfaces.
8) Tensor bundles on Riemann surfaces. The Riemann-Roch theorem (without
proof).
9) Teichm\uller space and moduli space for tori. Fundamental domain in the
Teichmuller space for tori.
10) Beltrami differentials on Riemann surfaces as generators of conformal
structures deformations. The tangent space to the moduli space.
11) Action of the Witt algebra (vector fields on the circle) at the moduli
space. Variations of the Riemann period matrix under the action of vector
fields.
12) Virasoro algebra action on the Dirac space associated to a Riemann
surface. The origin of the central charge.
transformations. Lie derivative of metric tensor with respect to a
generator of conformal transformation.
2) Mobius transformations of R^n. The Lie algebra of Mobius group.
Theorem: For n>=3 the Lie algebra of local conformal vector fields in R^n
is finite-dimensional and coincide with the Lie algebra of Mobius group.
3) Mobius transformations of R^n from isometries of R^{n+1,1}.
4) Weil, Schouten and Cotton tensors. The formula expressing the Riemann
tensor in terms of metric and Ricci tensors for n=3. Transformations
properties of the Weil and Cotton tensors under conformal changes of
Riemannian metric.
5) Necessary and sufficient conditions for conformal flatness (without
proof).
6) Isothermal coordinate on two-dimensional surfaces. Beltrami equation.
All 2-dimensional Riemannian manifolds are conformally flat. Local
conformal maps are holomorphic or anti-holomorphic maps.
7) Genus of hyperelliptic Riemann surfaces. Vector fields and
differentials on Riemann surfaces. Holomorphic differentials on
hyperelliptic Riemann surfaces.
8) Tensor bundles on Riemann surfaces. The Riemann-Roch theorem (without
proof).
9) Teichm\uller space and moduli space for tori. Fundamental domain in the
Teichmuller space for tori.
10) Beltrami differentials on Riemann surfaces as generators of conformal
structures deformations. The tangent space to the moduli space.
11) Action of the Witt algebra (vector fields on the circle) at the moduli
space. Variations of the Riemann period matrix under the action of vector
fields.
12) Virasoro algebra action on the Dirac space associated to a Riemann
surface. The origin of the central charge.