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1 Non-Euclidean Kime Manifold

So far, we considered spacekime as 5D Euclidean space where the 2D kime is a flat space. Does it make sense to generalize kime to non-Euclidean manifold and what is the rationale and meaning of a curved kime manifold?

We saw earlier the geometric representation of the flat spacekime using conical shapes. Let’s start with some visualization of the non-Euclidean 2D kime manifold.

## [1] 200 200
##           [,1]      [,2]      [,3]      [,4]      [,5]
## [1,] -3.141593 -3.110019 -3.078445 -3.046871 -3.015297
## [2,] -3.141593 -3.110019 -3.078445 -3.046871 -3.015297
## [3,] -3.141593 -3.110019 -3.078445 -3.046871 -3.015297
## [4,] -3.141593 -3.110019 -3.078445 -3.046871 -3.015297
## [5,] -3.141593 -3.110019 -3.078445 -3.046871 -3.015297
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,] 3.015297 3.046871 3.078445 3.110019 3.141593
## [2,] 3.015297 3.046871 3.078445 3.110019 3.141593
## [3,] 3.015297 3.046871 3.078445 3.110019 3.141593
## [4,] 3.015297 3.046871 3.078445 3.110019 3.141593
## [5,] 3.015297 3.046871 3.078445 3.110019 3.141593
## [1] 200 200
##      [,1]                                     
## [1,] " x:  0 \n r:  0 \n phi:  -3.142"        
## [2,] " x:  0.007 \n r:  0.084 \n phi:  -3.142"
## [3,] " x:  0.013 \n r:  0.118 \n phi:  -3.142"
## [4,] " x:  0.02 \n r:  0.144 \n phi:  -3.142" 
## [5,] " x:  0.026 \n r:  0.165 \n phi:  -3.142"
##      [,2]                                    
## [1,] " x:  0 \n r:  0 \n phi:  -3.11"        
## [2,] " x:  0.007 \n r:  0.084 \n phi:  -3.11"
## [3,] " x:  0.013 \n r:  0.118 \n phi:  -3.11"
## [4,] " x:  0.02 \n r:  0.144 \n phi:  -3.11" 
## [5,] " x:  0.026 \n r:  0.165 \n phi:  -3.11"