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Tangent Space

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Tangent Vector Motion along a trajectory is described by position and velocity. Position uses an originPosition uses an origin References the trajectoryReferences the trajectory Displacement points along the trajectory. Tangent to the trajectoryTangent to the trajectory Velocity is also tangentVelocity is also tangent x1x1 x2x2 x3x3

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Tangent Plane Motion may be constrained Configuration manifold QConfiguration manifold Q Velocities are not on the manifold.Velocities are not on the manifold. Set of all possible velocities Associate with a point x QAssociate with a point x Q N-dimensional set V nN-dimensional set V n Tangent plane or fiber T x Q x V nT x Q x V n V1V1 S1S1 S2S2 x V2V2

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Tangent Bundle Fibers can be associated with all points in a chart, and all charts in a manifold. This is a tangent bundle.This is a tangent bundle. Set is T Q Q V nSet is T Q Q V n Visualize for a 1-d manifold and 1-d vector.Visualize for a 1-d manifold and 1-d vector. V1V1 S1S1

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Twisted Bundles A tangent plane is independent of the coordinates. Coordinates are local to a neighborhood on a chart. Charts can align in different ways. Locally the same bundleLocally the same bundle Different manifold T QDifferent manifold T Q V1V1 S1S1

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Tangent Maps Map from tangent space back to original manifold. = T Q Q ; (x, v) (x) = T Q Q ; (x, v) (x) Projection map Projection map Map from one tangent space to another f : U W; U, W open f is differentiable T f : TU TW (x, v) (f(x), Df(x)v) Tangent map T f Df(x) is the derivative of f V1V1 S1S1

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Tangent Map Composition The tangent map of the composition of two maps is the composition of their tangent maps T f : TU TW; T g : TW TXT f : TU TW; T g : TW TX T( gf ) = T g T fT( gf ) = T g T f Equivalent to the chain rule next

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