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paweld
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I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).
A null geodesic is a path or curve in a 2 dimensional manifold that has a tangent vector with a zero magnitude at every point along the curve. This means that the path has no length and is essentially a straight line.
A regular geodesic is a path on a manifold that follows the shortest distance between two points. It is not constrained to have a zero tangent vector and can have a length greater than zero. A null geodesic, on the other hand, has a zero tangent vector and therefore has no length.
Null geodesics play a crucial role in understanding the geometry and topology of 2 dimensional manifolds. They are closely related to the concept of light rays in physics and can help determine the curvature and shape of the manifold. They are also important in theories of general relativity and cosmology.
Null geodesics can be calculated using the geodesic equation, which is a set of differential equations that describes the path of a geodesic on a manifold. In specific cases, such as in flat Euclidean space, the calculation can be simplified. In more complex cases, numerical methods may be used.
Yes, null geodesics can exist in higher dimensional manifolds. In fact, they are a fundamental concept in the study of higher dimensional geometry and can be used to characterize the properties of these manifolds. However, the equations and calculations may become more complex as the dimension of the manifold increases.