Basic concepts and results of Riemannian geometry

i=1 Q

\

j=1

{(x,y)∈V : f_{i j}(x,y) =0, g_{i j}(x,y)>0},

where the functions f_{i j},g_{i j}:V →Rare real-analytic, for all i=1,· · ·,p and j=1,· · ·,q.

Then, the set A is called subanalytic if each point ofRⁿadmits a neighborhood V⊂Rⁿ×R and B⊂Rⁿ×Ra bounded semianalytic subset such that A∩V ={x∈Rⁿ : (x,y)∈B}.

Finally, a function f :Rⁿ→Ris called subanalytic if its graph is a subanalytic subset of Rⁿ×R. It is worth to point out that subanalytic functions that is continuous when restricted to its closed domain satisfies the KŁ-property with desingularising function γ(t) = Dt^θ/θ with D >0 and θ∈ (0,1]; for more details see [20, Theorem 3.1]. For examples of subanalytic functions see e.g. [8,20,21].

2.2 Basic concepts and results of optimization on Hadamard manifolds

In this section, we recall some concepts, notations, and basics results about Riemannian manifolds and optimization. For more details see, for example, [32,59,62, 71]. Let us begin with concepts about Riemannian manifolds.

2.2.1 Basic concepts and results of Riemannian geometry

In this subsection we present some basic results about Riemannian geometry that will be used throughout Chapter6. For the next definitions, please see [32].

Definition 2.23 A differentiable manifold of dimension n is a set Mⁿ and a family of injective mappingsx_α:U_α ⊂Rⁿ→M of open sets U_αofRⁿinto M such that:

(i) ^S_αx_α(U_α) =M;

2.2 Basic concepts and results of optimization on Hadamard manifolds 27

(ii) for any pairα,β,withx_α(U_α)∩x_β(U_β) =W 6=∅,the setsx⁻¹_α (W)andx⁻¹

β (W) are open sets inRⁿand the mappingsx⁻¹

β ◦x⁻¹_α are differentiable;

(iii) The family{(U_α,x_α)}is maximal relative to the conditions(i)and(ii)above.

The indexnin the notationMⁿindicates the dimension ofM. When there is no confusion such index can be omitted. Unless when explicitly stated, through this sectionMdenotes a differentiable (which can be also a Riemannian) manifold of dimensionn.

Definition 2.24 Let M be a differentiable manifold. A differentiable function α : (−ε,ε)→M is called a (differentiable) curve in M. Suppose that α(0) = p∈M, and let

D

be the set of functions on M that are differentiable at p. The tangent vector to the curveαat t=0is a functionα⁰(0):

D

^→Rgiven by

α⁰(0)f = d(f◦α) dt

t=0

, f ∈

D

^.

A tangent vector at p is the tangent vector at t=0 of some curveα:(−ε,ε)→M with α(0) =p. The set of all tangent vectors to p is denoted by T_pM. The set T M={(p,v): p∈ M,v∈T_pM}is calledtangent budleof M.

Definition 2.25 Let M₁ⁿand M₂^mbe differentiable manifolds. A mapping F:M₁→M₂is differentiableat p∈M₁if given a parametrizationy:V ⊂R^m→M₂at F(p)there exists a parametrizationx:U ⊂Rⁿ→M₁at p such that F(x(U))⊂y(V)and the mapping

y⁻¹◦F◦x:U ⊂Rⁿ→R^m

is differentiable atx⁻¹(p). F is differentiable on an open set of M₁if F is differentiable at all of the points of this open set.

Definition 2.26 Avector fieldX on a differentiable manifold M is a correspondence that associates to each point p∈M a vector X(p)∈T_pM. The field is differentiable if the mapping X :M→T M is differentiable. The set of all vector fields on M of class C^∞ is denoted by

X

^(M).

Definition 2.27 A Riemanninan metric on a differentiable manifold M is a correspon-dence which associates to each point p of M an inner product (that is, a symmetric, bilin-ear, positive-defined form) hh·,·ii_p,on the tangent space T_pM.A differentiable manifold M with a given Riemannian metric is called aRiemannian manifold.

Definition 2.28 A differentiable mapping c:I→M of an open interval I⊂R into M a differentiable manifold M is called a parametrized curve. A vector field along a curve c :I → M is a differentiable mapping that associates to every t ∈I a tangent vector

2.2 Basic concepts and results of optimization on Hadamard manifolds 28

V(t)∈T_c(t)M. The vector field ^dc_dt is called thetangent vector fieldof c. The restriction of a curve c to a closed interval[a,b]⊂I is called asegment. If M is a Riemannian manifold, we define the lenght of a segment by

`^b_a(c) = Z _b

a

hhc⁰(t),c⁰(t)ii^1/2dt.

Definition 2.29 An affine connection ∇ on a differentiable manifold M is a mapping

∇:

X

^(M)×

X

^(M)→

X

(M), which is denoted by(X,Y)→^∇ ∇_XY and which satisfy:

(i) ∇_{f X+gY}Z= f∇_XZ+g∇_YZ (ii) ∇_X(Y+Z) =∇_XY+∇_XZ (iii) ∇_X(f Y) = f∇_XY+X(f)Y, for all X,Y,Z∈

X

^{(M)and f,g∈}

D

^(M).

Proposition 2.30 Let M be a differentiable manifold and let ∇be an affine connection on M. There exists an unique correspondence which associates to a vector field V along the differentiabla curve c:I →M another vector field ^DV_dt along c, called the covariant derivativeof V along c, such that:

(i) _dt^D(V+W) =^dV_dt +^dW_dt .

(ii) _dt^D(f V) =^{d f}_dtV+f^DV_dt ,where V is a vector field along c and f is a differentiable function on I.

(iii) If V(t) =Y(c(t)), then ^DV_dt =∇_dc/dtY.

Proof. See [32, p. 50].

Definition 2.31 Let M be a differentiable manifold and let∇be an affine connection on M. A vector field V along a curve c:I→M is called parallelif ^DV_dt =0,for all t∈I.

Proposition 2.32 Let M a differentiable manifold with an affine connection∇. Let c:I→ M be a differentiable curve in M and let V₀be a vector tangent to M at c(t₀), t₀∈I.Then there exist a unique parallel vector field V along c, such that V(t₀) =V (the field V(t)is called theparallel transportof V(t₀)along c).

Proof. See [32, p. 52].

Definition 2.33 A parametrized curve γ:I →M is a geodesic at t₀ ∈I if _dt^D dγ

dt

for t=t₀; ifγis a geodeic for all t ∈I, we say thatγis ageodesic. If[a,b]⊂I andγ:I→M is a geodesic, the restriction ofγto[a,b]is called ageodesic segmentjoiningγ(a)toγ(b).

2.2 Basic concepts and results of optimization on Hadamard manifolds 29

When there is no confusion, we will consider the notationP_q←p for the parallel transport along the geodesic segment γ joining p to q. Following [32, Chapter 3], let M be a Riemannian manifold. The exponential map exp_p :T_pM → M at p ∈M is defined by exp_p(v) =γ_v(1,p) for each v∈T_pM, where γ(·) =γ_v(·,p) is the geodesic starting at p with velocityv. Then exp_p(tv) =γ_v(t,p)for each real numbert.

Definition 2.34 Let M be a Riemannian manifold and p,q∈M.Consider the setΓp,q:=

{c:[a,b]→M | c is a piecewise differentiable curve joining p and q}. TheRiemannian distancefrom p to q is d(p,q):=inf{`(c)|c∈Γp,q}.

A Riemannian manifoldMiscompleteif the geodesics inMare defined for allt ∈R. Theorem 2.35 (Hopf-Rinow) Let M be a connected Riemannian manifold. Then the following conditions are equivalent:

(i) M is geodesically complete at a point p∈M.

(ii) M is geodesically complete, i.e., the geodesics in M are defined for all t∈R. (iii) For a fixed point p∈M, the set B[p,r]:={q∈M :d(p,q)≤r}is compact for

any r>0.

(iv) For any p∈M and any r>0, B[p,r]is compact.

(v) (M,d) is complete as a metric space. Namely, any Cauchy sequence of M is a convergent sequence.

Moreover, each one of above items(i)-(v)implies in the following:

(vi) For any two points p,q∈M there exists a geodesic (calledminimal geodesic)γ joining p to q such that`(γ) =d(p,q).

Proof. See [62, p. 84].

Definition 2.36 The curvature R of a Riemannian manifold M is a correspondence that associates to every pair X,Y ∈

X

^(M)a mapping R(X,Y):

X

^(M)→

X

^{(M)given by}

R(X,Y)Z=∇_Y∇_XZ−∇_X∇_YZ+∇_{(XY−Y X)}Z, Z∈

X

^(M),

where∇is the Riemannian connection of M.

Remark 2.37 The field XY−Y X ∈

X

^(M)is the unique vector field given by thebracket operation of X and Y . For more details about the bracket operation, see [32, Chapter 0, Section 5].

2.2 Basic concepts and results of optimization on Hadamard manifolds 30

Definition 2.38 Letσ⊂T_pM be a two-dimensional subspace of T_pM and let x,y∈σbe two linearly independent vectors. Then the sectional curvatureof M at p relative to the sectionσis given by

K(x,y) = hhR(x,y)x,yii_p p|x|²|y|²− hx,yi². The next definition can be found in [62, p. 222].

Definition 2.39 A complete simply connected Riemannian manifold M of nonpositive sectional curvature is called aHadamard manifold.

From now on in this work we will denote a Hadamard manifold by

M

. We remark that due to the Hadamard-Cartan’s Theorem [62, p. 222], if

M

is Hadamard, then the exponential map exp_p:T_p

M

^→

M

is a diffeomorphism for every p∈

M

^{and exp−1}p :

M

^→Tp

M

denotes its inverse. Denote byRthe real extended line, i.e.,R=R∪ {±∞}. Thedomain of a function f :

M

^→Ris denoted bydom(f):={p∈

M

^{: f(p)<}+∞}. Following [9]

and [71] we present below some concepts on optimization on Hadamard manifolds.

No documento On some boosted methods for DC programming and the extension of the DCA to Hadamard Manifolds (páginas 27-31)