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Exponential map - Definition and Overview |
| Related Words: Algorithmic, Cardinal, Decimal, Differential, Digital, Even, Figurate, Figurative, Finite, Imaginary, Infinite, Integral, Irrational, Logarithmic, Negative, Numeral, Odd, Ordinal |
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In differential geometry, the exponential map is the map from (a subset of) the tangent space <math>T_p M<math> of a Riemannian manifold M to M itself. It is defined in the following way:
For <math>v\in T_p M<math> there is a unique geodesic <math>\gamma^{}_v<math> such that <math>\gamma^{}_{}(0)=p<math> having a tangent vector <math>\gamma'(0)=v_{}^{}<math>.
Then <math>exp_p(v)=\gamma_v^{}(1).<math>
The name comes from the fact that it coincides with exponentiation of matrices in the case of bi-invariant metrics on Lie groups, when one is using a matrix representation of the group, and its Lie algebra as tangent space at the identity.
See also
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