No. You have several threads hinting that you are reading a text which presents manifolds in general through their embedding into a higher dimensional Euclidean space. The general definition of a manifold and its tangent space does not require this. I suggest that you pick up a reference where manifolds are treated properly.mertcan said:I think curvilinear coordinates generally define tangent space, but in curved space also defines normal component besides the tangent space. Am I right? I saw some close definition like this. Is it true?
Curved space is a concept in mathematics and physics that describes a space in which the laws of Euclidean geometry do not hold. It is a space that has a non-zero curvature, meaning that the shortest distance between two points is not a straight line. This concept is important in understanding the behavior of objects in the presence of gravity.
Curved space can affect our perception of the world in a number of ways. For example, in Einstein's theory of general relativity, gravity is described as the curvature of space-time caused by the presence of massive objects. This means that the path of objects, such as planets, is affected by the curvature of space, leading to phenomena such as the bending of light and the creation of black holes.
Curvilinear coordinates are a system of coordinates used to describe points in a curved space. They are different from the Cartesian coordinates we are used to, in which points are described using x, y, and z coordinates. Curvilinear coordinates use different variables, such as r, θ, and φ, to describe points in a curved space, and they are useful for solving problems in physics and engineering that involve curved spaces.
Curvilinear coordinates are closely related to curved space because they are used to describe points in a curved space. The choice of curvilinear coordinates depends on the specific curvature of the space in question. By using curvilinear coordinates, we can calculate distances, angles, and other properties of objects in a curved space, which would be difficult or impossible to do using Cartesian coordinates.
It is difficult to visualize curved space because our brains are used to thinking in terms of Euclidean geometry. However, we can use analogies and mathematical models to help us understand the concept. For example, we can imagine the surface of a sphere as a curved space, where the shortest distance between two points is a great circle. We can also use mathematical tools, such as differential geometry, to describe and visualize curved spaces.