# 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$L(γ)= \int_a^b \sqrt{g(\dot{γ}(t),\dot{γ}(t))} dt$.

The

**Riemannian distance**on a connected Riemannian manifold (M, g) is$\text{dist} : \mathcal{M} × \mathcal{M} → \mathbb{R} : \text{dist}(x, y) = \inf_Γ L(γ)$

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$\langle \text{grad}f (x), ξ \rangle _x = \text{D}f (x) [ξ] , ∀ξ \in T_xM.$

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.

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