# What is curvature?

## What Does curvature Mean

The Latin word curvatūra came to our language as curvature . The concept refers to the condition of curved (bent or crooked). The idea of ​​curvature is also used with respect to the deviation that a curved line has from a straight line.

For example: "Criminals tried to take advantage of the curvature of the wall to hide, but were discovered" , "Poor posture can lead, in the long run, the curvature of the spine vertebral" , "The curvature of the screen shocked the public ” .
If someone talks about the curvature of a television, to cite one case, they are referring to the fact that its screen is not straight. The curvature of a cell phone (mobile), meanwhile, is linked to its curved edges. In these cases, the curvature can represent both an aesthetic and a functional aspect, or a fusion of both. Regardless of the purpose of this feature in an appliance, electronic device or car, among other products, fashion trends make it inevitable that its duration is limited, so sooner or later the curvature is replaced by the angular edges, and vice versa.

In geometry and mathematics, curvature can be the magnitude or number that measures this quality. It is, in this framework, the amount that a geometric object deviates from a line or a plane.
The notion of curvature of space-time derives from the theory of general relativity , which postulates that gravity is an effect of the curved geometry that space-time has. According to this theory, bodies that are in a gravitational field make a curved path in space. The curvature of space-time is measured according to the so-called curvature tensor or Riemann tensor .
The displacement curvature , on the other hand, is a theory stating that a vehicle could move at a speed greater than the speed of light from a distortion that generates a greater curvature in space-time.
There is a quantity called radius of curvature that is used to measure the curvature of an object belonging to geometry as if it were a surface, a curved line or, more generally, a differentiable manifold found in Euclidean space .
If we take an object or a curved line as a reference, its radius of curvature is a geometric magnitude that we can define at each of its points, and it is equivalent to the inverse of the absolute value of the curvature in all of them. We must not forget that the curvature is the alteration that goes through the direction of the tangent vector to a given curve as we move along it.

One of the measurements that we can make on a given surface is the Gaussian curvature , a number belonging to the set of reals that represents the intrinsic curvature for each of the regular points. It is possible to calculate it starting from the determinants of the two fundamental forms of the surface.
The first fundamental form of the surface is a 2-covariant tensor that presents symmetry and is defined in the tangent space to each of its points; it is the metric tensor (that is, rank 2, used to define concepts such as volume, angle and distance) that the Euclidean metric induces on the surface. The second, on the other hand, is the projection of the covariant derivative that is carried out on the normal vector to the surface, and is induced by the first fundamental form.
Gaussian curvature is generally different at each point on the surface and is related to its principal curvatures. The sphere is a special case of a surface, since in all its points it has the same curvature.

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