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umerfarooque
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Can anyone explain the concept of manifold and Reimanni manifold in plain language ?? And what are its applications??
Thanks
Thanks
How so? The riemannian metric allows you to calculate the arc lengths of differentiable curves; this is not the same thing that a metric equipped to a metric space does. What notion of distance are you alluding to?saminator910 said:The Riemann metric gives the family of inner products at a point, and in turn gives you the notion of distance on the manifold.
A manifold is a mathematical object that is used to describe spaces that are curved, such as the surface of a sphere. It is a generalization of the concept of a plane or a space, and can have any number of dimensions.
A Riemannian manifold is a type of manifold that is equipped with a Riemannian metric, which is a way of measuring distances and angles on the manifold. This metric allows for the calculation of geometric properties such as curvature and volume.
Manifolds are important in mathematics because they provide a way to study and understand curved spaces, which are often encountered in physics and other natural sciences. They also have applications in areas such as differential geometry, topology, and dynamical systems.
Manifolds are different from Euclidean spaces in that they do not have a constant curvature or a well-defined notion of distance and angle. Instead, the properties of a manifold vary from point to point, and the geometry is described by a set of rules that apply locally.
Some examples of manifolds include the surface of a sphere, a torus, a cylinder, and the space we live in (which is a 3-dimensional manifold). Other examples can be found in nature, such as the shape of a DNA molecule or the trajectory of a planet around the sun.