ATMW Riemannian Geometry (2012)

This will be a 2-week event in which first 10 days will be devoted to rigorous instructional lectures in foundational topics in Riemannian Geometry. Last 2 days will have a conference component in which the various invited speakers will give overview lectures on their topic of research interest and mention some possible open problems for the PhD students.

Brief description: This will be a 2-week event in which first 10 days will be devoted to rigorous instructional lectures in foundational topics in Riemanenian Geometry; this will also include a series of lectures by Professor G´rard Besson, Institut Fourier, Grenoble, France, on his joint work with Courtois and Gallot on the minimal entropy theorem. Last 2 days will have a conference component in which the various invited speakers will give overview lectures on their topic of research interest and mention some possible open
problems for the PhD students.

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Lecturewise break-up of the syllabus
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• Lecture 1: Quick review of Gauss’s theory of surfaces.
• Lecture 2: Smooth manifolds - definition and examples.
• Lecture 3: Smooth functions, bump functions (smooth urysohn lemma);tangents vectors, vector fields, tensor fields - definition and properties (of tensors).
• Lecture 4: Covariant derivative, connection and parallel transport.
• Lecture 5: Metric tensor, Riemannian manifolds, Levi-Civita connection, curvature tensors and curvatures.
• Lecture 6: Lengths of curves, distance function, first variation of arc length and geodesics.
• Lecture 7: Exponential map, Jacobi fields, Gauss lemma.
• Lecture 8: Hopf-Rinow theorem.
• Lecture 9: Second variation of arc length and the Index form.
• Lecture 10: Cartan-Hadamard and Bonnet-Myers theorems.
• Lecture 11: Riemannian immersions, second fundamental form, Gauss equation.
• Lecture 12: Models of constant curvature; Cartan’s theorem on the determination of the metric by (constant) curvatures.
• Lecture 13: Riemannian submersions; O’Neill’s submersion equation.
• Lectures 14 and 15: Klingenberg’s injectivity radius estimate, Synge’s theorem.
• Lecture 16: Rauch and Toponogov comparison theorems (include proof of Rauch but only state Toponogov).
• Lecture 17: Bishop and Bishop-Gromov volume comparison theorems.
• Lectures 18, 19, 20 and 21: Preissmann’s theorem, Eberlein-O’Neill compactification, Busemann functions, classification of isometries (intoelliptic, parabolic and axial or hyperbolic).
• Lectures 22, 23, 24 and 25: Symmetric spaces of compact and noncompact type; their curvatures.
• Lectures 26 to 29 will be by G´rard Besson.

The list of speakers in the workshop includes Ravi Aithal, C S Aravinda, Anandateertha Mangasuli, Akhil Ranjan, Harish Seshadri.

Workshop Schedule

Speakers in the Workshop:

RA : Ravi Aithal

Department of Mathematics
University of Mumbai, Mumbai

CSA : C S Aravinda
TIFR Centre for Applicable Mathematics
Bangalore

AM : Anandateertha Mangasuli
Department of Mathematics
IISER, Bhopal

AR : Akhil Ranjan
Department of Mathematics
IITB, Mumbai

HS : Harish Seshadri
Department of Mathematics
IISc, Bangalore

Tutors: Vikram Aithal, CBS, Mumbai
Atreyee Bhattacharya and Soma Maity IISc, Bangalore

List of Selected applicants (Deadline for confirmation (by email) of participation: 30th June, 2012.)

Outside Bangalore:

How to reach TIFR CAM, Bangalore

* From Bangalore Internation Airport (Distance 28km)

You can hire a City taxi or private taxi from Airport to TIFR CAM, Yelahanka new town.
Landmark : Yelahanka new town Dairy Circle

* From Bangalore City Railway Station or Majestic Bus Stand (Distance 16km)

You can hire a City Taxi or Auto Rickshaw and ask drop for Yelahanka new town. The fare for City Taxi will be approximately Rs.400. For Auto Rickshaw fare will be approximately Rs.200.

You can also use BMTC Bus service to reach TIFR CAM. You can board into following buses which comes to Yelahanka new town bus stand.

Buses to Yelahanka new town : 402B, 402D, 284D

TIFR CAM is located walkable distance from Yelahanka new town bus stand.

You may also visit the link below http://math.tifrbng.res.in/contactus
for links to the route maps.