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.