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What is Differential Geometry?

Differential Geometry is the study of geometric properties using Differential and Integral Calculus. It is a branch of mathematics dealing with geometrical forms and the intrinsic properties of curves and surfaces as related to differential calculus and mathematical analysis.

Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.

The general idea of natural equations for obtaining curves from local curvature appears to have been first considered by Leonhard Euler in 1736, and many examples with fairly simple behavior were studied in the 1800s.

When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge’s paper in 1795, and especially, with Gauss’s publication of his article, titled ‘Disquisitiones Generales Circa Superficies Curvas’, in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827.

Source: Wikipedia

What are Geometric Properties?

Geometric Properties are properties that solely depend on the geometry of an object, not of how
it happens to appear in space. These are properties that do not change under congruence. They can be derived from the geometry of a solid body or particle.

What is the curvature of a curve?

Well, a line is not curved at all; its curvature has to be zero. A circle with a small radius is more ”curved” than a circle with a large radius. Circles and lines have constant curvature. Curves that are not (pieces of) circles or lines will have a curvature varying from point to point.

In elementary geometry, one meets a lot of examples of curves: straight lines, circles, conic sections, cubic curves, graphs of functions defined on an interval or the whole real line, intersections of surfaces etc.

1-dimensional topological manifolds with boundary. They have a finite or countable number
of open connected components and each connected component is homeomorphic
either to an open, closed, or half-closed interval or to a circle. The class of 1-dimensional manifolds with boundary is wider than the class of simple arcs, it includes much more important examples of curves, but as it fixes the local structure of a curve quite strictly, it excludes examples of
curves having certain kind of singularities. For example, figure-eight shaped curves, like Bernoulli’s lemniscate, are not topological manifolds, because the self-intersection point in the middle does not have a neighborhood with the required property.

Algebraic plane curves may have a finite number of singular points, for example, self-intersections, so they are not necessarily 1-dimensional manifolds, but removing the singular points, the remaining set is a 1-dimensional manifold, maybe empty. On the other hand, algebraic curves are very specific curves. For example, if a straight line intersects an algebraic curve in an infinite
number of points, then it is contained in the curve. In particular, the graphs periodic non-constant functions (like the sine function) are not algebraic curves.

Source: http://etananyag.ttk.elte.hu/FiLeS/downloads/_01_Csikos_Differential_geometry.pdf

What is a curve?

Wikipedia: “In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight.”
Most of the time, the curves are described by means of their Cartesian equation: for example y-2 x = 1.

So we can think of a 2D-curve as a set of points:
C = {(x, y) ∈ R2|F(x, y) = c},  for a real c and a function F : R2 –> R.

For a 3D-curve, we need a pair of equations,

F1(x, y, z) = c1
F2(x, y, z) = c2

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