Differential geometry is an important aspect in the filed of mathematics in which we combine the study of calculus, both integral and differential equations to solve problems of geometry.
It also includes the topics under the category of differential topology. Although the study of differential geometry doesn’t always include differential calculus, it can also include the topics of linear algebra too

Differential geometry includes the theory of surfaces, curves or planes in any 3 dimensional Euclidean spaces. Differential Geometry helps to find out the geometrical interpretation of the figures in both space and at a given point of time.
Differential Geometry includes the study of structure of curves, surfaces, motions that are non rigid, the study of curvilinear trajectories, curvature of curve, curvature of surface, and many more.
We generally use the concept of curves for studying differential geometry rather than studying the specific points, because all the boundary conditions on the curved surfaces are either original boundaries or act as some constraints.
Consider the following curve drawn on the top of a circle as shown below:

     Differential Geometry Curve 
 
Here, the curve is represented by C(u), where the point C (u0) is the value of  the point at u0, on the curve, and can be denoted by p = C (u0). Similarly, C (u1) is value of the point on the curve when the point u0 is moved to the point u1, on the curve. T denotes the tangent to the curve; N (T) denotes the normal to curve at the point T, and N (u0) and N (u1) are the corresponding normal to the point C (u0) and C (u1).
Now, given any curve, it can be parametrized by considering the following important terms:
  •     Finding the length of an arc of a curve, this is denoted by C (u).
  •     Finding the tangent of a curve, this is denoted by C‘(u) = T = Cu / || Cu ||, where Cu =  $\frac{\partial C(u)}{\partial u}$
  •     Finding the normal of any curve, this is denoted by C ‘‘(u) = N = [Cuu – (T * Cuu) T] /( || Cuu – (T * Cuu) T ||), where, Cuu =  $\frac{\partial^{2}C(u)}{\partial u^{2}}$
  •    Finding the binormal of any curve, this is denoted by B = (Cuu * Cuu) / || Cuu * Cuu ||,
  •     Finding the curvature of any curve, this is denoted by k = - T * N (T), where N(T) is N (u) $\frac{\partial u}{\partial s}$  and T is equal to Cu $\frac{\partial u}{\partial s}$, which on further computation will give the value (– Cu * Nu) / (Cu * Cu), which can again calculated in norm form as k = || Cu  * Cuu || / || Cu || ^ 3.
Similarly, we have the above mentioned terms in case of surfaces also, as shown below:
 
       Curve Curvature 
Here, the surface is represented as S (u, v), p is any point on the surface, as was in the case of curve, we have p = S (u0, v0), and T is the plane of tangents Su and Sv.
Differential Geometry is widely applied in the filed of almost every research and science areas, whether it’s in the field of physics, economics, and engineering, statistics, and computer sciences or in the field of communication through wireless devices. In the field of medical sciences too, the concept of differential geometry is used for the study of different genetic structures.

We know that, in the field of econometrics, many econometrics models take the form of geometrical figures, which are known as manifolds, especially the exponential curves.
In the filed of mathematics, the concept to tangent space is widely applied. In the field of statistics, the concept of metric and general tensors is applied.
There is a huge connection between the filed of information theory and differential geometry, in connection with the problems relating to the parameterization’s choices, which uses the concept of affine connections. This topic of affine connections of the subject differential geometry is widely applied in the manifolds of statistics, the projections, in the field of problems of inferences.In the field of biology, the concept of Frenet Frame is widely used in finding out the structures of various molecular reactions. Similarly, the concept of discrete form of Frenet frame is widely used in mathematical geometrical figures.

 For example: consider the following discrete form of Frenet Frame for a curve which is drawn along a cube.

    Curve Along a Cube 
 
From the above figure, we can find out the tangents, normal and binormal at any given point j, thus helping in the correct visualization of the object.

Differential geometry is widely applied in the study of various polymers, in the field of chemistry too, where we use the famous formula of Eyring’s Formula which is also deducted from the discrete form of Frenet Frame.

As we know there are two fundamental forms in the study of differential geometry of surfaces which are as follows:
1). The First fundamental form of a surface: It is denoted by Is and is calculated by finding the metric of the given surface, hence, Is = T. T.
2).  The Second fundamental form of a surface: It is denoted by IIs and is calculated as IIs = - T . N(T).
These are widely applied to analyze the different forms of curvature of a given curve or surface. With the help of the two fundamental forms of a surface, we are able to derive an operator, W, which is known as the Weingarten Operator which is calculated as follows:
W = (Is $^{ -1}$) IIs.
This operator is applied widely, as every Weingarten surface gives rise to the integral two manifolds.
The following is a list of some problems of differential geometry, which are given along with their solutions too. These are as follows:

Problem 1: Write the Frenet Frame formulas which are used for representing the derivatives of the tangent to a curve, the normal and bi normal of a curve, which are parametrized by their usual length of an arc.
Solution: Let T, N, and B denotes the usual tangent, normal, and bi normal of a given curve respectively,
Then, the Frenet Frame formulas are written as follows:
T ‘ (s) = k (s) * N (s)
N ‘ (s) = - k (s) * T (s) – t (s) * B (s)
B ‘ (s) = t (s) * N (s).
Here, k(s) denotes the curvature and t (s) denotes the torsion. 

Problem 2: Given a point u0 of a smooth surface. Then find the condition for the point u0 to be umbilical?
Solution: Firstly, Let the point u0 is umbilical. Let K be the Gaussian curvature and H be the mean curvature.
Now, the point u0 will be umbilical if and only if the principal curvatures Kand K2 will be equal to each other. Hence,
K = K1 * K2 =  K1 *  K1 =  K12 and
H = (K1 + K2) / 2 = (K1 + K1) / 2 =  K1.
Combining both the equations we get, K = H2.

Now conversely, let H2 = K, then, 
K1 * K2  = (K1 + K2 / 2) 2
After eliminating K1 * K2 from both the sides, after simplification, we will get, 
0 = (K1 – K2 / 2) 2, this equation would hold true if and only if K1 = K2.

Hence, the condition for the point u0 to be umbilical is as follows:
H2 = K.
Hence, we have proved that if H2 = K, then the principal curvatures are equal and vice versa.

Problem 3: Given a coordinate value as S (u, v) = (u, u2 + v2, - v), then find the normal N of a unit normal vectors considering the above coordinates?

Solution: Firstly, we will find the tangent vectors (by finding the first derivatives of the given surface) to the given surface which is:
Tu = (1 , 2u, 0)
Tv = (0, 2v, -1).
Hence, to find the unit normal vector we will find from the formula as mentioned below: C ‘‘(u) = N = [Cuu – (T * Cuu) T] / || Cuu – (T * Cuu) T ||, where, Cuu = $\frac{\partial^{2}C(u)}{\partial u^{2}}$  .
Hence, N = (- 2u, 1, 2v) / || (- 2u, 1, 2v) ||,
This would give the three coordinates of the normal as:
((- 2u / sqrt of ( 4 u2 + 4 v22 + 1); 1 / sqrt of ( 4 u2 + 4 v2 + 1); 2v / sqrt of ( 4 u2 + 4 v2 + 1)), which is the answer.
Differential Geometry has the following important elements which form the basic for studying the elementary differential geometry, these are as follows:

Length of an arc: This is the total distance between the two given points, made by an arc of a curve or a surface, denoted by C (u) as shown below:

 Length of an Arc 

Tangent to a curve: The tangent to a curve C (u) is the first partial derivative of the curve at a fixed given point u and is denoted by C ‘(u) or its also denotes as a ‘ (s), where the curve is represented by a (s), as shown below:

  Tangent of a Curve
Hence, a ‘(s) or C ‘ (u) or T are the similar notations used for denoting tangent to a curve.

Curvature: Curvature is defined as any direction in any normal degree, (denoted as n) into its side which is empty or blank. In mathematics, we can find the curvature of any surface or curve by calculating the ratio of the rate of change of the angle made by the tangent that is moving towards a given arc to the rate of change of the its arc length, that is, we can define a curvature as follows:
C ‘’ (s) or a’’(s) = k (s) n (s), where k (s) is the curvature, which can be understood better by looking at the following diagram:
    Curvature 
We can now prove that if a’(s) * a ‘(s) = 1, then this would definitely imply that:
a’’(s) * a ‘(s) = 0. 
Thus a curvature is basically the capability of changing of a curve form a ‘ (s) to a ‘ (s + $\Delta$ s) in a given direction as shown below:

   Change in Curvature
 
Once, we have calculated the tangent T to a given cure, its easy to find out the value of normal N and binormal B of a given curve, which gives us the elements of a famous formula in differential geometry, which is known as Frenet Frames, which is a function of F (s) = (T(s), N (s), B(s)), where C (s) is any given curve in the space.

For example: consider the following diagram of a circle, and then we can differentiate its various elements as follows:
.   Derivative of Circle Elements 
 
  • The length of the arc, s is given to be equal to r * $\theta$, this implies, $\theta$ = s/ r, whose coordinates will be as follows:
a (s) = [r cos (s/r), r sin (s/r)],
  • The tangent would be calculated by taking the first partial differentiation of a (s), which would be:
    a ‘ (s) or T = [-  sin (s/r), cos (s/r)],
and
  •  The curvature of the circle would by calculated by taking the second partial differentiation of a (s) as shown below:
a '' (s) or k = [-  cos (s/r) / r, - sin (s/r) / r ], which will give k = - 1 / r2 a (s), on further calculation, thus, mod of k would be equal to 1 / r.

The course of Differential Geometry can be better understood by reading the below mentioned summarized notes:
The concept of Curve in Differential Geometry: Any curve can be represented by C (u) at a point u = uo, which can be further examined for its parametrization, by depicting its length of the arc, its tangent, normal and bi normal. These derivatives if combined together makes jointly an important formula known as The Frenet Serret Formula, because they gives us the properties related to the kinematic field of any curve or surface in a three dimensional space. 
As we know, that differential geometry is basically the concept which is widely applied to find out the dimensions in any moving images. In a one dimensional space, we find the differential geometry of a curve, which is calculated by finding its curvature and torsion along its curve.

Torsion of a curve: is the rate of change of the curve’s plane which is osculating as shown below:

     Torsion of a Curve 
 
We can see from the above diagram that the whole plane is moving in a particular direction, which is termed as Torsion and is denoted as t.
Torsion t is positive if the plane goes in the direction of positive x axis and it becomes negative if the plane goes in the negative direction of the x axis. t is calculated as t = - N * B’. Note that 1 / t is known as the radius of Torsion and is denoted by phi.

Curvature and its Lines: The principal direction (PD) of any curvature is that direction which is the resulting amount of the maximum and the minimum of a normal curvature.
There are basically two kinds of principal curvature that can be possessed by any given curve. These are:
  • The Gaussian Curve: This principal curvature is denoted by K, where K = K1 * K2. Note that K1 and K2 are the principal curvatures, where a principal curvature is defined as the maximum and the minimum of the normal curvature.
  • The Mean Curve: This kind of principal curvature is denoted by H, which is the average mean of two curvatures, as H = ( K1 + K2)  / 2
By the definition of curvature, we can conclude that:
  •     The principal curvature of parabolic point will be K = 0
  •     The principal curvature of elliptical point will be K > 0
  •     The principal curvature of hyperbolic point will be K < 0.
These principal curvatures are applied widely in case of the mapping of tangents.