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Tangent Space

Tangent Space
There are several possible approaches to generalizing the notion of a directional derivative Df(x)[η] = lim t→0  f(x+tη)−f(x) / t to a real-valued function f deﬁned on a manifold.
A ﬁrst possibility is to view η as a derivation at x.
A second, perhaps more intuitive approach to generalizing the directional derivative is to replace t → (x+tη) by a smooth curve γ on M through x (i.e., γ(0) = x).
Let M be a manifold. A smooth mapping γ : R → M : t → γ(t) is termed a curve in M.
A tangent vector ξx to a manifold M at a point x is a mapping from Fx(M) to R such that there exists a curve γ on M with γ(0) = x, satisfying ξxf = `γ(0)f := d(f(γ(t))) / dt | t=0, for all f ∈ Fx(M).
Fx(M) denote the set of smooth real-valued functions defined on a neighborhood of x.
The point x is called the foot of the tangent vector ξx. We will often omit the subscript indicating the foot and simply write ξ for ξx.
Such a curve γ is said to realize the tangent vector ξx.
Given a tangent vector ξ to M at x, there are infinitely many curves γ that realize ξ.
The tangent space to M at x, denoted by TxM, is the set of all tangent vectors to M at x.
This set admits a structure of vector space.
In the same way that the derivative of a real-valued function provides a local linear approximation of the function, the tangent space TxM provides a local vector space approximation of the manifold.
Later we can deﬁne mappings, called retractions, between M and TxM, which can be used to locally transform an optimization problem on the manifold M into an optimization problem on the more friendly vector space TxM.
Example tangent spaces
The tangent space to St(p, n) at X: TxSt(p, n) = {Z ∈ R^n×p : X^T Z + Z^T X = 0}.
The tangent space to quotient spaces: vertical space and horizontal space.
Riemannian Geometry
We further need a notion of length that applies to tangent vectors.
A manifold whose tangent spaces are endowed with a smoothly varying inner product is called a Riemannian manifold.
The smoothly varying inner product is called the Riemannian metric.
Strictly speaking, a Riemannian manifold is thus a couple (M, g), where M is a manifold and g is a Riemannian metric on M.
A vector space endowed with an inner product is a particular Riemannian manifold called Euclidean space.
The length of a curve γ : [a, b] → M on a Riemannian manifold (M, g) is deﬁned by
​L(γ)=∫abg(γ˙(t),γ˙(t))dtL(γ)= \int_a^b \sqrt{g(\dot{γ}(t),\dot{γ}(t))} dtL(γ)=∫ab​g(γ˙​(t),γ˙​(t))​dt
.
The Riemannian distance on a connected Riemannian manifold (M, g) is
​dist:M×M→R:dist(x,y)=inf⁡ΓL(γ)\text{dist} : \mathcal{M} × \mathcal{M} → \mathbb{R} : \text{dist}(x, y) = \inf_Γ L(γ) dist:M×M→R:dist(x,y)=infΓ​L(γ)
​
where Γ is the set of all curves in M joining points x and y.
Metrics and Riemannian metrics should not be confused. A metric is an abstraction of the notion of distance, whereas a Riemannian metric is an inner product on tangent spaces. There is a link since any Riemannian metric induces a distance, the Riemannian distance.
Given a smooth scalar ﬁeld f on a Riemannian manifold M, the gradient of f at x, denoted by grad f(x), is deﬁned as the unique element of TxM that satisﬁes
​⟨gradf(x),ξ⟩x=Df(x)[ξ],∀ξ∈TxM.\langle \text{grad}f (x), ξ \rangle _x = \text{D}f (x) [ξ] , ∀ξ \in T_xM.⟨gradf(x),ξ⟩x​=Df(x)[ξ],∀ξ∈Tx​M.
​
The gradient of a function has the following remarkable steepest-ascent properties:
The direction of grad f(x) is the steepest-ascent direction of f at x.
The norm of grad f(x) gives the steepest slope of f at x.
Quotient Manifolds
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