Exponential Map

On a Lie Group, exp is a Map from the Lie Algebra to its Lie Group. If you think of the LieAlgebra as the Tangent Space to the identity of the Lie Group, exp() is defined to be , where isthe unique Lie Group Homeomorphism from the Real Numbers to the Lie Group such thatits velocity at time 0 is .

On a Riemannian Manifold, exp is a Map from the Tangent Bundle of the Manifold to theManifold, and exp() is defined to be , where is the unique Geodesic traveling through thebase-point of such that its velocity at time 0 is .

The three notions of exp (exp from Complex Analysis, exp from Lie Groups, and exp from Riemanniangeometry) are all linked together, the strongest link being between the Lie Groups and Riemannian geometrydefinition. If is a compact Lie Group, it admits a left and right invariant Riemannian Metric. With respectto that metric, the two exp maps agree on their common domain. In other words, one-parameter subgroups are geodesics. In thecase of the Manifold , the Circle, if we think of the tangent space to 1 as being theImaginary axis (y-Axis) in the Complex Plane, then

• Friend
• Friendly Giant Group
• Friendly Pair
• Frieze Pattern
• Frobenius-König Theorem
• Frobenius Map
• Frobenius Method
• Frobenius-Peron Equation
• Frobenius Pseudoprime
• Frobenius Theorem
• Frobenius Triangle Identities
• Frontier
• Frullani's Integral
• Frustum
• Fréchet Bounds
• Fréchet Derivative
• Fréchet Space
• Fubini Principle
• Fuchsian System
• Fuchs's Theorem
• Fuhrmann Circle
• Fuhrmann's Theorem
• Fuhrmann Triangle
• Fuller Dome
• Full Reptend Prime