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trees and plants

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The length of ##∇_{T}T## is pretty much the same thing but with a factor of the curvature of the curve thrown in. So, in a sense, it measures the degree to which the curve is not "straight" (i.e. geodesic).To define the curvature of a manifold in two dimensions, you can use the same concept of geodesic curvature mentioned above. This is a local property of the manifold and does not depend on any embedding in a higher dimensional space. The fact that it is defined in terms of the covariant derivative makes it independent of the coordinate system used.As for what helped mathematicians define

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trees and plants

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Ibix

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A circle embedded in a plane has extrinsic curvature, but it has no intrinsic curvature, and it is the latter type of curvature that relativity uses. Generally, you seem to be thinking of extrinsic curvature, the idea that a space is curved relative to a space in which it is embedded. We have no evidence that spacetime is embedded in anything. The curvature it has is intrinsic curvature which can be defined and measured without reference to an external space.universe function said:Curvature can be informally defined as the deviation from a straight line in the context of curves, a circle in R^2 has curvature

No. A geodesic is the path with extremal distance between two events. And it is not a curve, it is by definition a straight line. The projection of the geodesic on to a subspace (such as the 3d spatial path of an object following a geodesic in 4d spacetime) may be curved in that subspace. The length-squared of a geodesic may be negative in Lorentzian manifolds, but I don't think it makes sense to talk of a negative length. That's just a positive length with a needless sign convention.universe function said:Another topic is about geodesics and their length, a geodesic informally could be defined as the shortest distance between two points in a geometric space like a surface,it could be a curve.

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trees and plants

This is the definition in the context of general relativity. I am talking here about differentiable riemannian manifolds.The negative length of a geodesic should have applications to be more interested in it or not?Ibix said:A circle embedded in a plane has extrinsic curvature, but it has no intrinsic curvature, and it is the latter type of curvature that relativity uses. Generally, you seem to be thinking of extrinsic curvature, the idea that a space is curved relative to a space in which it is embedded. We have no evidence that spacetime is embedded in anything. The curvature it has is intrinsic curvature which can be defined and measured without reference to an external space.

No. A geodesic is the path with extremal distance between two events. And it is not a curve, it is by definition a straight line. The projection of the geodesic on to a subspace (such as the 3d spatial path of an object following a geodesic in 4d spacetime) may be curved in that subspace. The length-squared of a geodesic may be negative in Lorentzian manifolds, but I don't think it makes sense to talk of a negative length. That's just a positive length with a needless sign convention.

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The length of a spacetime interval may be positive and negative, corresponsing to spacelike and timelike separation.universe function said:This is the definition in the context of general relativity. I am talking here about differentiable riemannian manifolds.The negative length of a geodesic should have applications to be more interested in it or not?

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trees and plants

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ChinleShale

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trees and plants

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https://www.preposterousuniverse.com/grnotes/universe function said:

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ChinleShale

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Yes but things get a little complicated. At a point where the curvature of the curve is not zero, the covariant derivative of the unit normal vector ##∇_{T}N## has a non-zero component parallel to the curve. The length of this component is just the curvature back again.universe function said:

If one ignores this component and just takes the length of the component that is perpendicular to the plane spanned by ##T## and ##N## then one gets the tendency of the curve to move out of that plane. This is called the torsion of the curve.

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ChinleShale

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Here is the calculation of the covariant derivative of the unit normal to the curve. Try doing the calculation for the unit torsion vector.

##0=T⋅<T,N>=<∇_{T}T,N> +<T,∇_{T}N> = ##κ + ##<T,∇_{T}N>## so the component of the covariant derivative of ##N## parallel to ##T## is negative the curvature times ##T##. For the length of the component perpendicular to the plane spanned by ##T## and ##N## one just calls it torsion. So ##∇_{T}N = ##-κ##T + τB##.

The generalization of curvature and geodesics in higher dimensions is important because it allows us to understand the behavior of space in more complex and realistic scenarios. In everyday life, we experience three dimensions, but in fields such as physics and mathematics, higher dimensions are often used to describe the behavior of particles and objects. By generalizing curvature and geodesics, we can better understand the behavior of space in these higher dimensions.

In higher dimensions, curvature is defined as the measure of how much a space curves or bends at a given point. This can be visualized as the amount of deviation from a straight line that occurs when moving along a path in the space. In three dimensions, we can think of curvature as the bending of a sheet of paper, while in higher dimensions, it becomes more complex and difficult to imagine.

Geodesics in higher dimensions can be thought of as the shortest path between two points in a curved space. For example, on a sphere, the geodesic would be the shortest path between two points on the surface, which would be a segment of a great circle. In four dimensions, we can think of the geodesic as the path that a particle would follow in the curved space-time around a massive object, such as a planet or star.

The generalization of curvature and geodesics in higher dimensions has a significant impact on our understanding of the universe. It allows us to better understand the behavior of space-time in scenarios such as black holes, where the curvature of space-time is extremely high. It also helps us to develop more accurate models and theories in fields such as cosmology and string theory.

Yes, there are practical applications of the generalization of curvature and geodesics in higher dimensions. One example is in the field of robotics, where understanding the behavior of space in higher dimensions is crucial for designing and programming robots to navigate through complex environments. Additionally, the generalization of curvature and geodesics has applications in fields such as computer graphics and computer vision.

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