Riemannian Geometry and General Relativity

Riemannian geometry is designed to describe the universe of creatures who live on a curved surface or in a curved space and do not know about the world of higher dimensions or do not have any access to it.

One of the main notions of the Riemannian geometry is the notion of connection, which is, in fact, the key notion of the entire geometry, though it is not always explicitly formulated. The connection (or parallel transport) allows to compare what is happening at two distant points of a curved space, in spite of the fact that there is no direct and immediate way to communicate between these points. Recently connections appeared in the theory of gauge fields which is considered a basis of the modern physics of elementary particles. Earlier, in the 1910's, A.Einstein discovered that the Riemannian geometry can be successfully used to describe General Relativity which is in fact a classical theory of gravitation. (Here the word ``classical" stands as opposite to ``quantum", but the quantum theory of gravitation is still terra incognita!)

By its intrinsic beauty, as well as by wealth of applications the Riemannian geometry lies at the core of modern mathematics.

The subjects covered in the course are as follows:

• 1. Manifolds, Riemannian metrics.
• 2. Affine connections and Riemannian connections.
• 3. Geodesics.
• 4. Curvature tensor, Ricci curvature and scalar curvature.
• 5. Comparison theorems.
• 6. Einstein's law of gravitation. The gravitational red shift.
• 7. The Schwarzschild solution. Black holes.
• 8. Einstein equations in the presence of matter and electromagnetic field.

Note:

Home Assignment 1

Solutions to Home Assignment 1

Home Assignment 2

Solutions to Home Assignment 2

Home Assignment 3

Solutions to Home Assignment 3

Home Assignment 4

Solutions to Home Assignment 4

Textbooks:

1. Riemannian Geometry, by Manfredo Perdigão do Carmo. Birkhäuser, Boston, 1993.

2. General Theory of Relativity, by P.A.M.Dirac. Princeton University Press, 1996.

The textbooks complement one another. The first one (M.P. do Carmo) is mathematical, and it is not large, very well written, self-contained and has a good set of exercises. The second one (P.A.M. Dirac) is a physics classic, brilliantly written, very short (69 pages) and cheap. It is also self-contained but from a physicist's point of view. Reading both books (which sometimes describe the same topics but quite differently in style) you will have a chance to compare mathematical and physical approaches, which should lead to a deeper understanding.

Office: 460 Lake Hall. Phone: Ext.5676 E-mail: shubin@neu.edu

Class meetings: Tuesday and Thursday 7:15 -- 8:45 p.m.

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