- Oggetto:
Submanifolds and Holonomy
- Oggetto:
Submanifolds and Holonomy
- Oggetto:
Academic year 2016/2017
- Teacher
- Prof. Carlos Olmos (Lecturer)
- Year
- 1° anno
- Type
- A scelta dello studente
- Credits/Recognition
- 6 CFU
- Course disciplinary sector (SSD)
- MAT/03 - geometria
- Delivery
- Tradizionale
- Language
- Inglese
- Attendance
- Obbligatoria
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Sommario del corso
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Program
Course: Submanifolds and Holonomy (30 hours).
Both graduate and undergraduate students are admitted
(for the last type of students the exams will be simpler)
For approving the course:
-
A written exam at the middle of the course (not obligatory).
-
A written exam at the end (students who passed the first exam would have to do a simpler exam).
Topics.
-
Euclidean submanifolds (brief account).
-
Basic facts and formulae.
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Isoparametric submanifolds and its focal manifolds.
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Homogeneity of isoparametric submanifolds (Thorbergsson’s Theorem)
-
Polar representations and s-representations
-
Normal Holonomy.
-
The Rank rigidity theorem for submanifolds.
-
Riemannian geometry (brief account).
-
Basic facts.
-
Homogeneous spaces.
-
Naturally reductive spaces.
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Holonomy.
The following is the main part of the course and is reasonable self-contained.
This is for the convenience of the students who are not particularly interested
in submanifold geometry.
-
The Berger-Simons Holonomy Theorem.
- Holonomy systems.
- The Simons Holonomy Theorem (a geometric proof)
- The Berger Holonomy Theorem (a geometric proof)
-
The Skew-Torsion Holonomy Theorem
- Fixed point sets of isometries and homogeneous submanifolds.
- Naturally reductive spaces.
- Totally skew 1-forms with values in a Lie algebra.
- The derived 2-formwithvalues in a Lie algebra.
- The Skew-Torsion Holonomy Theorem.
- Applications to naturally reductive spaces.
- The full isometry group of naturally reductive spaces.
- The holonomy of naturally reductive spaces.
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Suggested readings and bibliography
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Main bibliography.
Berndt, J., Console, S., and Olmos, C., Submanifolds and Holonomy, Second Edition, Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL (to appear in March 2016).
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