# Course of Differential Geometry

Written by: Ruslan Sharipov

Written by Ruslan Sharipov, this online differential geometry textbook is available in PDF, PostScript and other formats. The author describes the text as a “first acquaintance with the differential geometry.”

1. CURVES IN THREE-DIMENSIONAL SPACE
1. Curves. Methods of defining a curve. Regular and singular points
of a curve
2. The length integral and the natural parametrization of a curve
3. Frenet frame. The dynamics of Frenet frame. Curvature and torsion of a spacial curve
4. The curvature center and the curvature radius of a spacial curve.
5. The evolute and the evolvent of a curve
6. Curves as trajectories of material points in mechanics
2. ELEMENTS OF VECTORIAL AND TENSORIAL ANALYSIs
1. Vectorial and tensorial fields in the space
2. Tensor product and contraction
3. The algebra of tensor fields
4. Symmetrization and alternation
5. Differentiation of tensor fields
6. The metric tensor and the volume pseudotensor
7. The properties of pseudotensors
8. A note on the orientation
9. Raising and lowering indices
10. Gradient, divergency and rotor. Some identities of the vectorial analysis
11. Potential and vorticular vector fields
3. CURVILINEAR COORDINATES
1. Some examples of curvilinear coordinate systems
2. Moving frame of a curvilinear coordinate system
3. Change of curvilinear coordinates
4. Vectorial and tensorial fields in curvilinear coordinates
5. Differentiation of tensor fields in curvilinear coordinates
6. Transformation of the connection components under a change of a coordinate system
7. Concordance of metric and connection. Another formula for Christoffel symbols
8. Parallel translation. The equation of a straight line in curvilinear coordinates
9. Some calculations in polar, cylindrical, and spherical coordinates
4. GEOMETRY OF SURFACES
1. Parametric surfaces. Curvilinear coordinates on a surface
2. Change of curvilinear coordinates on a surface
3. The metric tensor and the area tensor
4. Moving frame of a surface. Veingarten’s derivational formulas
5. Christoffel symbols and the second quadratic form
6. Covariant differentiation of inner tensorial fields of a surface
7. Concordance of metric and connection on a surface
8. Curvature tensor
9. Gauss equation and Peterson-Codazzi equation
5. CURVES ON SURFACES
1. Parametric equations of a curve on a surface
2. Geodesic and normal curvatures of a curve
3. Extremal property of geodesic lines
4. Inner parallel translation on a surface
5. Integration on surfaces. Green’s formula
6. Gauss-Bonnet theorem

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Course of Differential Geometry 