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.”

- CURVES IN THREE-DIMENSIONAL SPACE
- Curves. Methods of defining a curve. Regular and singular points

of a curve - The length integral and the natural parametrization of a curve
- Frenet frame. The dynamics of Frenet frame. Curvature and torsion of a spacial curve
- The curvature center and the curvature radius of a spacial curve.
- The evolute and the evolvent of a curve
- Curves as trajectories of material points in mechanics

- Curves. Methods of defining a curve. Regular and singular points
- ELEMENTS OF VECTORIAL AND TENSORIAL ANALYSIs
- Vectorial and tensorial fields in the space
- Tensor product and contraction
- The algebra of tensor fields
- Symmetrization and alternation
- Differentiation of tensor fields
- The metric tensor and the volume pseudotensor
- The properties of pseudotensors
- A note on the orientation
- Raising and lowering indices
- Gradient, divergency and rotor. Some identities of the vectorial analysis
- Potential and vorticular vector fields

- CURVILINEAR COORDINATES
- Some examples of curvilinear coordinate systems
- Moving frame of a curvilinear coordinate system
- Change of curvilinear coordinates
- Vectorial and tensorial fields in curvilinear coordinates
- Differentiation of tensor fields in curvilinear coordinates
- Transformation of the connection components under a change of a coordinate system
- Concordance of metric and connection. Another formula for Christoffel symbols
- Parallel translation. The equation of a straight line in curvilinear coordinates
- Some calculations in polar, cylindrical, and spherical coordinates

- GEOMETRY OF SURFACES
- Parametric surfaces. Curvilinear coordinates on a surface
- Change of curvilinear coordinates on a surface
- The metric tensor and the area tensor
- Moving frame of a surface. Veingarten’s derivational formulas
- Christoffel symbols and the second quadratic form
- Covariant differentiation of inner tensorial fields of a surface
- Concordance of metric and connection on a surface
- Curvature tensor
- Gauss equation and Peterson-Codazzi equation

- CURVES ON SURFACES
- Parametric equations of a curve on a surface
- Geodesic and normal curvatures of a curve
- Extremal property of geodesic lines
- Inner parallel translation on a surface
- Integration on surfaces. Green’s formula
- Gauss-Bonnet theorem

View this Free Online Material at the source:

Course of Differential Geometry