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This question reminds me of when I was a graduate student. At some point Gelfand asked me "What is the rank of a symmetric space" and I just spit spat back the usual definition, something like what Matrix found in Wikipedia. Gelfand shook his head as if I had said something really stupid and proceeded to explain:

Euclidean space, hyperbolic space, complex projective space (and so on) are rank one. Why? Because if you have two pairs of points and the distance between them is the same, then there is an isometry that takes one pair of points to the other. ONE invariant is all you need to determine whether two pairs of points are the same up to isometries.

The Grassmannian of two-planes in ${\mathbb R}^4$ has rank two : you need two invariants to determine if two pairs of points are equivalent up to isometry. Take two planes in four-space passing through the origin. Draw a circle with center zero in one plane. Project it orthogonally onto the second plane. You get an ellipse, but you cannot compare it to the circle because it lives on a different plane so project it back to the first plane. The minor and major axes of your ellipse (with respect to the circle) are two invariants that are preserved by any isometry of the pair of planes. Conversely if you have two pairs of planes that have the same two invariants, then there is an isometry of the Grassmannian that takes one pair of planes to the other.

I went back home and the uninsightful book I was reading on symmetric spaces went back to the library the next day.

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This question reminds me of when I was a graduate student. At some point Gelfand asked me "What is the rank of a symmetric space" and I just spit back the usual definition"is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero" as , something like what Matrix found in Wikipedia. Gelfand shook his head as if had said something really stupid and proceeded to explain:

Euclidean space, hyperbolic space, complex projective space (and so on) are rank one. Why? Because if you have two pairs of points and the distance between them is the same, then there is an isometry that takes one pair of points to the other. ONE invariant is all you need to determine whether two pairs of points are the same up to isometries.

The Grassmannian of two-planes in ${\mathbb R}^4$ has rank two : you need two invariants to determine if two pairs of points are equivalent up to isometry. Take two planes in four-space passing through the origin. Draw a circle with center zero in one plane. Project it orthogonally onto the second plane. You get an ellipse, but you cannot compare it to the circle because it lives on a different plane so project it back to the first plane. The minor and major axes of your ellipse (with respect to the circle) are two invariants that are preserved by any isometry of the pair of planes. Conversely if you have two pairs of planes that have the same two invariants, then there is an isometry of the Grassmannian that takes one pair of planes to the other.

I went back home and the uninsightful book I was reading on symmetric spaces went back to the library the next day.

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This question reminds me of when I was a graduate student. At some point Gelfand asked me "What is the rank of a symmetric space" and I just spit back the usual definition "is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero" as Matrix found in Wikipedia. Gelfand shook his head as if had said something really stupid and proceeded to explain:

Euclidean space, hyperbolic space, complex projective space (and so on) are rank one. Why? Because if you have two pairs of points and the distance between them is the same, then there is an isometry that takes one pair of points to the other. ONE invariant is all you need to determine whether two pairs of points are the same up to isometries.

The Grassmannian of two-planes in ${\mathbb R}^4$ has rank two : you need two invariants to determine if two pairs of points are equivalent up to isometry. Take two planes in four-space passing through the origin. Draw a circle with center zero in one plane. Project it orthogonally onto the second plane. You get an ellipse, but you cannot compare it to the circle because it lives on a different plane so project it back to the first plane. The minor and major axes of your ellipse (with respect to the circle) are two invariants that are preserved by any isometry of the pair of planes. Conversely if you have two pairs of planes that have the same two invariants, then there is an isometry of the Grassmannian that takes one pair of planes to the other.

I went back home and the uninsightful book I was reading on symmetric spaces went back to the library the next day.