Differential Geometry: A beginner's journey.

A good summary:

http://www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

Focusing on surface:

http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces?action=render

Applications of Differential Geometries:

Helping to solve other unsolved maths problem:

http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

Possible applications of differential geometry concepts:

1. Elementary Geometry

2. Center of an arc determined with straightedge and compass.
3. Surface areas: Circle, trapezoid, triangle, sphere, frustum, cylinder, cone...
4. Power of a point  with respect to a circle.
5. Euler's line  goes through the orthocenter, the centroid and the circumcenter.
6. Euler's circle  is tangent to the incircle and the excircles  (Feuerbach, 1822).
7. Barycentric coordinates & trilinears  examplify  homogeneous coordinates.
8. Elliptic arc: Length of the arc of an ellipse between two points.
9. Perimeter of an ellipse. Exact formulas and simple ones.
   • Circumference of an ellipse: Unabridged discussion.
10. Surface of an ellipse.
11. Volume of an ellipsoid  (either a spheroid or a scalene ellipsoid).
12. Surface area of a spheroid  (oblate or prolate ellipsoid of revolution).
    • Surface area of a general ellipsoid.
13. Quadratic equations in the plane describe ellipses, parabolas, or hyperbolas.
14. Centroid of a circular segment. Find it with Guldin's (Pappus) theorem.
15. Parabolic arc of given extremities  with a prescribed apex between them.
16. Focal point of a parabola. y = x ² / 4f (where f is the focal distance).
17. Parabolic telescope: The path from infinity to focus is constant.
18. Make a cube go through a hole in a smaller cube.
19. Octagon: The relation between side and diameter.
20. Constructible regular polygons  and constructible angles (Gauss).
21. Areas of regular polygons of unit side: General formula & special cases.
22. For a regular polygon of given perimeter, the more sides the larger the area.
23. Curves of constant width: Reuleaux Triangle and generalizations.
24. Irregular curves of constant width. With or without any circular arcs.
25. Solids of constant width. The three-dimensional case.
26. Constant width in higher dimensions.
27. Fourth dimension. Difficult to visualize, but easy to consider.
28. Volume of a hypersphere and hyper-surface area, in any dimensionality.
29. Hexahedra. The cube is not the only polyhedron with 6 faces.
    • Unabridged discussion.
30. Descartes-Euler Formula: F-E+V=2 but restrictions apply.

    Topology:

31. Metric spaces:  The motivation behind more general  topological  spaces.
32. Abstract topological spaces  are defined by calling some subsets  open.
33. Closed sets  are sets (of a topological space) whose complements are open.
34. Subspace F of E:  Its open sets are the intersections with F of open sets of E.
35. Separation axioms:  Flavors of topological spaces, according to  Trennung.
36. Compactness of a topological space:  Any open cover has a  finite subcover.
37. Real-valued continuous functions on compact sets  attain their extremes.
38. Borel sets.  Tribes  form the topological foundation for  measure theory.
39. Locally compact sets  contain a  compact neighborhood  of every point.
40. General properties of sequences  characterize topological properties.
41. Continuous functions  let the  inverse image  of any open set be open.
42. Restrictions remain continuous.  Continuous extensions may be impossible.
43. The product topology  makes projections continuous on a cartesian product.
    • Tychonoff's Theorem:  Any product of compact spaces is compact.
44. Connected sets  can't be split by open sets.  The empty set  is  connected.
45. Path-connected sets  are a special case of  connected sets.
46. Homeomorphic sets.  An  homeomorphism  is a  bicontinuous function.
47. Arc-connected spaces are path-connected.  The converse need not be true.
48. Homotopy:  A progessive transformation of a  function  into another.
49. The fundamental group:  The homotopy classes of all loops through a point.
50. Homology and Cohomology.  Poincaré duality.
51. Descartes-Euler Formula:  F-E+V = 2, but restrictions apply.
52. Euler Characteristic:   c   (chi)  extended beyond its traditional definition...
53. Winding number  of a continuous planar curve about an outside point.
    • The  dog on a leash theorem.
    • A topological  proof of the fundamental theorem of algebra.
54. Fixed-point theorems  by  Brouwer,  Shauder  and  Tychonoff.
55. Turning number  of a planar curve with a well-defined oriented tangent.
56. Real projective plane  and Boy's surface.
57. Hadwiger's  additive continuous functions of d-dimensional rigid bodies.
58. Eversion of the sphere.  An homotopy  can  turn a sphere inside out.
59. Classification of surfaces:  "Zero Irrelevancy Proof" (ZIP) by J.H. Conway.
60. Braid groups:  Strands, braids and pure braids.

    Completeness:

61. Complete metric space:  A space in which all Cauchy sequences converge.
62. Flawed alternatives to completeness.
63. Banach spaces  are complete normed vector spaces.
64. Fréchet spaces  are generalized Banach spaces.

    Fractal Geometry:

65. Fractional exponents  were first conceived by Nicole d'Oresme (c. 1360).
66. The von Koch curve (and  snowflake):  Dimension of self-similar objects.
67. Hausdorff dimension is revealed by a covering with balls of radius  < e.
68. The Julia set and the Fatou set of an analytic function  are complementary.
69. The Mandelbrot set was so named by  Adrien Douady  &  John H. Hubbard.

    Angles and Solid Angles:

70. Planar angles  (from one direction to another)  are  signed  quantities.
71. Bearing:  Unless otherwise specified, this is the angle  west of north.
72. Solid angles  are to spherical patches what planar angles are to circular arcs.
73. Circular measures:  Angles and solid angles aren't quite dimensionless...
74. Solid angle formed by a trihedron :   Van Oosterom & Strackee  (1983).
75. Solid angle subtended by a rhombus.  Apex of a right  rhombic pyramid.
76. Formulas for solid angles  subtended by patches with simple shapes.
77. Right ascension and declination.  Precession of celestial coordinates  (a,d).

    Curvature and Torsion:

78. Curvature of a planar curve:  Variation of inclination with distance  dj/ds.
79. Curvature and torsion  of a three-dimensional curve.
80. Distinct curvatures and  geodesic  torsion  of a curve drawn on a surface.
81. The two fundamental quadratic forms  at a point of a parametrized surface.
82. Lines of curvature:  Their  normal  curvature is extremal at every point.
83. Geodesic lines.  Least length is achieved with  zero geodesic curvature.
84. Meusnier's theorem:  Tangent lines have the same  normal curvature.
85. Gaussian curvature of a surface.  The  Gauss-Bonnet theorem.
86. Parallel-transport of a vector around a loop.  Holonomic angle of a loop.
87. Total curvature of a curve.  The Fary-Milnor theorem for knotted curves.
88. Linearly independent components  of the  Riemann curvature tensor.

    Planar Curves:

89. Cartesian equation of a straight line:  passing through two given points.
90. Confocal Conics:  Ellipses and hyperbolae sharing the same pair of  foci.
91. Spiral of Archimedes:  Paper on a roll, or groove on a vinyl record.
92. Hyperbolic spiral:  The inverse of the  Archimedean spiral.
93. Catenary:  The shape of a thin chain under its own weight.
94. Witch of Agnesi.  How the versiera (Agnesi's cubic) got a weird name.
95. Folium of Descartes.
96. Lemniscate of Bernoulli:  A quartic curve shaped like the  infinity symbol.
97. Along a Cassini oval, the product of the distances to the two foci is constant.
98. Limaçons of Pascal:  The cardioid  (unit epicycloid) is a special case.
99. On a Cartesian oval, the weighted average distance to two poles is constant.
100. The envelope of a family of curves  is everywhere tangent to one of them.
101. The evolute of a curve  is the locus of its centers of curvature.
102. Involute of a curve:  Trajectory of a point of a line  rolling  on that curve.
103. Parallel curves  share their normals, along which their distance is constant.
104. The nephroid  (or  two-cusped epicycloid )  is a  catacaustic  of a circle.
105. Freeth's nephroid:  A special  strophoid  of a circle.
106. Bézier curves  are algebraic splines.  The cubic type is the most popular.
107. Piecewise circular curves:  The traditional way to specify curved forms.
108. Intrinsic equation  [curvature as a function of arc length]  may have  spikes.
109. The quadratrix (trisectrix) of Hippias squares the circle and trisects angles.
110. The parabola  is  constructible  with straightedge and compass.
111. Mohr-Mascheroni constructions  use the compass alone  (no straightedge).

Collection of Mathematica articles:

http://demonstrations.wolfram.com/siteindex.html