note-math

Probably not the perfect motivation

Restricting the metric to yields metric-manifold

Intuitively, in Euclidean , we can "rotate", and the composition of rotations corresponds to the addition of "angles"

The latter should be the of 's Killing-field as a one-parameter homomorphism to 's isometry

Calculate with geodesic. Calculate geodesic with, for example, stereographic projection coordinates. For the geodesic starting at , the result is denoted as [trigonometric-function] trigonometric function . The power series expansion of at can be calculated using inverse function theorem

Homomorphism is reflected in, according to power series

Now, there is multiplication on , expressed as the addition of angles

can be decomposed into distance and direction ,

Multiplication in is defined as the multiplication of distances in and the multiplication of directions in or the addition of angles

The multiplicative inverse of is represented as the additive inverse of the angle

The multiplicative inverse of is the inverse of the distance in and the inverse of the direction in

Distributive law

• The distributive law for distance multiplication in is that of , meaning that scaling after vector addition is equal to vector addition after scaling
• The meaning of the distributive law for direction multiplication in is that rotation is a linear map, and rotating after vector addition is equal to vector addition after rotation

Unlike and , because and , it can be said that itself is capable of multiplication

Algebra or , called complex numbers

is the multiplication of elements of length in , and also the multiplication that preserves the length of

or or

Complex conjugation means that the distance remains unchanged but the direction is reversed or