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In summary, the metric tensor is a key component in describing the geometry of a manifold, regardless of the coordinate system used. It can determine whether a geometry is Euclidean, Hyperbolic, or Elliptic by examining the sum of angles in a triangle. Parallelism in Riemannian geometry is based on geodesics, which can also be determined using the metric tensor.

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maajdl

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In the same way the metric has different representations for different systems of coordinates, but it still represents the same thing / geometry.

Indeed the metric is all that is needed to describe the geometry, if it is Euclidean for example.

The way to check if -for example- it is Euclidean, doesn't depend on the system of coordinates which is used.

For example, the total curvature will always be the same independently of the coordinate system used to calculate it.

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maajdl

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"Though not sure about this because different coordinate systems on the same manifold can lead to different metrics!"

I was meaning that the information about the geometry that is contained in the metric tensor is independent of the coordinates used. (that's the meaning of the metric being a tensor)

Regarding the notion of parallelism, you need first to define what a line is, since parallelism, as far as I know, is related to "lines". In the context of Riemannian geometry (ie based on a metric), parallelism is defined on the basis of geodesics. The existence of parallel geodesics can be verified by exploring the metric tensor.

If the distinction between Euclidean, Hyperbolic of Elliptic geometry is defined by comparing the sum of the angles of a triangle to Pi, then the metric tensor is definitively able to detect this distinction. This is because, 1) the metric tensor is all that you need to define parallel transport and "lines" and 2) it defines the geometry in the tangent space (distances and angles).

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blue_raver22

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In terms of parallel lines, the metric plays a crucial role. In Euclidean geometry, parallel lines are defined as lines that never intersect and have the same slope. This can be represented mathematically using the Euclidean metric, which measures distances in a straight line. In hyperbolic geometry, the metric is different, and parallel lines are defined as lines that never intersect but have a different slope. This leads to the famous result that in hyperbolic geometry, the sum of the angles in a triangle is less than 180 degrees.

In conclusion, the metric is indeed the key to understanding the geometry of a space. By examining the metric, we can determine whether a space is Euclidean, hyperbolic, or elliptic. This has important implications for many fields of science, including cosmology, where the curvature of space is a crucial factor in understanding the universe.

The metric system is a decimal-based system of measurement used in most countries around the world. It is based on seven base units: meter (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), mole (amount of substance), and candela (luminous intensity).

The metric system is used because it is a more consistent and logical system of measurement compared to the traditional Imperial system. It is also easier to convert between units and is used by the majority of countries, making it a universal system for trade and communication.

The metric system does not directly affect the existence of parallel lines. Parallel lines exist in Euclidean geometry, which is based on axioms and postulates, not on a specific system of measurement. However, the metric system can be used to measure the distance between parallel lines.

No, parallel lines can never intersect. By definition, parallel lines are two lines that are always the same distance apart and never meet. If two lines do intersect, they are not parallel.

Two lines can be proven parallel if they have the same slope and do not intersect. This can be done through various methods, such as using the slope-intercept form of a line or using the properties of alternate interior angles or corresponding angles.

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