Symmetries in Geometry

Symmetries in Geometry ´ Cartan From Euclid to Elie

Claudiu C. Remsing Geometry, Groups and Control (GGC) research group Department of Mathematics, Rhodes University

First Mini-Symposium on Geometry, Groups, and Control Grahamstown, December 6, 2018

C.C. Remsing (Rhodes University)

Symmetries in Geometry

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Outline Symmetry A few slogans A few questions Symmetry in physics Symmetry in mathematics

Symmetries in geometry Euclid: shape and size Gauss: non-Euclidean geometry Galois: group of permutations Lie: continuous groups of transformations Klein: groups in classical geometries; the “Erlanger Programm” Riemann: the metric tensor Cartan: generalized spaces

Homogeneous spaces: a few ideas C.C. Remsing (Rhodes University)


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Symmetry: A Few Slogans Slogan One

(Hermann Weyl, 1952)

“Beauty is bound up with symmetry.”

Slogan Two

(Marcus du Sautoy, 2008)

“Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication.”

Slogan Three

(Ian Stewart, 2013)

“Symmetry is an immensely important concept.” C.C. Remsing (Rhodes University)


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Symmetry: A Few Questions

(1/3)

Question

(Q1)

Where is symmetry ?

Short answer Almost everywhere.

Origins of symmetry Symmetries can be found in art (visual arts; music) culture; religious and secular symbols are often symmetric the natural world science (biology; chemistry; physics; computer science) mathematics C.C. Remsing (Rhodes University)


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(2/3)

Question

(Q2)

What is symmetry ?

Short answer Symmetry is change without change.

Commentary symmetry (of a mathematical structure): a transformation of that structure, of a specified kind, that leaves specified properties of the structure unchanged



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(3/3)

Question

(Q3)

Why is symmetry important ?

Short answer Symmetry is a concept that is simultaneously obvious and profound.

Commentary evolution has programmed us to be oversensitive to symmetry (Symmetry in the undergrowth is either someone about to eat you or something you could eat.) our brains seem to be hard-wired to find meaning in symmetry C.C. Remsing (Rhodes University)


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Symmetry in Physics

(1/2)

A reasonable history of physics can be given in terms of ever-expanding place for symmetry in understanding the physical world. A (physical) object has symmetry if it appears the same when viewed from a different perspective. Physicists have generalized the term “symmetry” from descriptions of (physical) objects to descriptions of laws of nature.

Galileo Galilei Isaac Newton

(1564–1642) (1643–1727)

Galilean invariance: the laws of motion are invariant with respect to an observation in an inertial frame of reference the laws of classical physics are invariant with respect to the Galilean group C.C. Remsing (Rhodes University)


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Symmetry in Physics

(2/2)

Albert Einstein

(1879–1955)

special relativity: the laws of nature are the same even when the frame of reference is moving close to the speed of light general relativity: the laws of nature are invariant even when acceleration is taken into account (i.e., the observer is accelerating)

Emmy Noether

(1882–1935)

for every (continuous) symmetry of the laws of physics, there must exist a related conservation law for every conservation law, there must exist a related (continuous) symmetry

Hermann Weyl

(1885–1955)

gauge invariance (it refers to a type of local symmetry) C.C. Remsing (Rhodes University)


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Symmetry in Mathematics: A Few More Slogans Slogan Four

(1/2)

(Anthony Zee, 1986)

“Symmetry and mathematics are closely related. Structures heavy with symmetry would also naturally be rich in mathematics.”

Slogan Five

(D.V. Alekseevskij et al., 1988)

“The simplicity of an object under investigation is in many cases equivalent to its symmetry.”

Slogan Six

(N.S. Yanofsky – M. Zelcer, 2017)

“Symmetry lies at the heart of the nature of mathematics.” C.C. Remsing (Rhodes University)


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Symmetry in Mathematics

(2/2)

Category theory and the symmetries of mathematics symmetries: are discovered when we find that seemingly different mathematical phenomena are really in the same category as an already known transformation (and are thereby subsumed under a larger domain) category theory: grew out of the unification of (certain) topological and algebraic phenomena

Slogan: Final

(F. William Lawvere, 1964)

“Thus we seem to have partially demonstrated that, even in foundations, not substance but invariant form is the carrier of the relevant mathematical information.” C.C. Remsing (Rhodes University)


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(1/2)

Space A space consists of points. (In a space we shall be able to identify points.) The points cannot be distinguished from each other. (The points are equivalent under transformations of the space.)

A few approaches to space manifold: description by coordinates charts (i.e., by locally relating to a model space) scheme: description through local functions (with points corresponding to maximal ideals in function spaces) assembling a space from local pieces (e.g., simplices, cells) so that the properties of the space are reduced to the combinatorics of this assembly pattern C.C. Remsing (Rhodes University)


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(2/2)

(Geometric) structure A (geometric) structure on the space consists of constraints on its transformations.

Differential geometry

(M, σ)

differentiable manifold: a principle for identifying points and (tangent) vectors across different local representations tensor field (tensor calculus): coordinate representations of geometric objects and the transformations of those representations under coordinate change connection: a scheme for the comparison of the infinitesimal geometries at different points C.C. Remsing (Rhodes University)


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Euclid: Shape and Size Euclid (Eukleides) of Alexandria

(325–265 bc)

Euclid: the “father of geometry” “Elements of Geometry”:

makes geometry into a deductive science

shape (similarity) and size (congruence): “equal” figures are not distinguished (in terms of their properties) congruence: two figures are equal (or congruent) if there exists an isometry (i.e., a distance-preserving transformation) sending one figure onto the other the notion of equality of figures needs to be meaningful: reflexivity; symmetry; transitivity (cf. the group axioms !) (Associating a transformation group to the geometry is to fix a criterion of identity for geometric objects in the geometry) C.C. Remsing (Rhodes University)


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Galois: Group of Permutations

´ Evariste Galois

(1811–1832)

simplicity of an equation (algebraic or differential) means the possibility of solving it in some concrete form (a proper degree of) symmetry

⇒

integrability

concept of group (of permutations) “Galois theory”: the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial



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Gauss: Non-Euclidean Geometry Carl Friedrich Gauss Nikolai Ivanovich Lobachevsky János Bolyai

(1777–1855) (1792–1856) (1802–1860)

“I am becoming more and more convinced that the necessity of our [Euclidean] geometry cannot be proved, at least not by human reason nor for human reason.” (Gauss, 1817) non-Euclidean geometry: 1829)

first published account (Lobachevsky,

non-Euclidean geometry:

second published account (Bolyai, 1831)

hyperbolic geometry: the pseudosphere (and all other surfaces of constant negative curvature) provides a local model for hyperbolic geometry (Beltrami, 1868)



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Riemann: The Metric Tensor Bernhard Riemann

(1826–1866)

“Theory of Surfaces” (1827): Gauss develops the geometry on a surface based on its (first) fundamental form. “On the Hypotheses which underlie Geometry” (Göttingen, 1854): As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. geometry: space (manifold) equipped with a field quantity (Riemannian metric) (A field of geometric quantities of (order 1 and) type W is a GL (n, R)-equivariant map σ : R1 (M) → W, from the manifold of coframes of M to a GL (n, R)-manifold.)



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Lie: Continuous Groups of Transformations

Sophus Lie

(1842–1899)

1873: Lie started examining partial differential equations, hoping he could find a theory which was analogue to the Galois theory of (polynomial) equations continuous transformation groups (Lie groups); infinitesimal groups (Lie algebras) “Theory of Transformation Groups” (co-authored with Friedrich Engel): I (1988); II (1890); III (1893)



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Klein: Groups in Classical Geometries

Felix Klein

(1849–1925)

all the classical geometries (of the 19th century) can be represented as homogeneous spaces most classical geometries are part of projective geometry isomorphic geometries; subgeometries; classification of geometries



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Klein: What is Geometry ? The “Erlanger Programm”

(1872)

geometry: the study of the properties of a space that are invariant under a group of transformations “Given a manifoldness and a group of transformations of the same; to investigate the configurations belonging to the manifoldness with regard to such properties as are not altered by the transformations of the group.” (Klein) (A) a geometry could (and should) be studied by studying the group of transformations that preserve the geometry (B) a group which acts (transitively) on a space determines a geometry



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Cartan: Generalized Spaces

´ Cartan Elie

(1869–1951)

“espaces généralisés”: Cartan’s generalized spaces include both Klein’s homogeneous spaces and Riemann’s local geometry (1935) (Cartan) geometry: bundle


the study of a connection on a principal fibre


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Homogeneous Spaces: A Few Ideas

(1/12)

Definition: (smooth) homogeneous space

(G, M)

M is a connected (smooth) manifold G is a Lie group the group G acts (smoothly and) transitively on M: θ : G × M → M,

(g , x) 7→ g · x

1

1·x =x

2

g2 · (g1 · x) = (g2 g1 ) · x

3

for any x, y ∈ M, there exists g ∈ G such that g · x = y


(x ∈ M) (x ∈ M, g1 , g2 ∈ G)


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(2/12)

Basic assumptions

(G, M)

G is connected the group G acts effectively on M: G 3 g 7→ θg := θ(g , ·) ∈ Diff (M) is one-to-one (hence, G can be viewed as a subgroup of Diff (M))



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Homogeneous Spaces: A Few Ideas Examples

(3/12) (G, M)

Lie groups (a Lie group is a homogeneous space in several ways) spheres:

Sn = SO(n + 1)/SO(n)

projective spaces: RPn = SO(n + 1)/O(n) Grassmann manifolds: Gr(k, n) = SO(n)/S(O(k) × O(n − k)) (k-dimensional subspaces in Rn ) Stiefel manifolds: k-frames in Rn )

V(k, n) = SO(n)/SO(n − k)

(orthonormal

(some of) the spaces associated with classical geometries: En (Euclidean), Hn (hyperbolic), An (affine), Ln = R1,n−1 (Lorentzian)



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(4/12)

Archetypal example: coset manifolds

(G, G/H)

If G is a Lie group and H is a closed Lie subgroup, then the orbit set G/H has a (unique) smooth manifold structure such that the quotient map π : G → G/H is a submersion; the map (natural action) λ : G × G/H 3 (g , aH) 7→ g · (aH) := (ga)H ∈ G/H is a transitive (Lie group) action.



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(5/12)

Characterization of homogeneous spaces Let (G, M) be a homogeneous space, and let o ∈ M be any point. Then the isotropy subgroup (of o) H = Go := {g ∈ G : g · o = o} is a closed Lie subgroup of G; the map F : G/H → M,

g H 7→ g · o

is an equivariant diffeomorphism (i.e., a diffeomorphism such that F ◦ λg = θg ◦ F for all g ∈ G.)



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Homogeneous Spaces: A Few Ideas Remark 1

(6/12) (G, M),

H = Go ≤ G

The equivariant diffeomorphism F : G/H → M carries the natural action of G on G/H to the action of G on M: F ◦ λg = θg ◦ F for all g ∈ G. (This means that the actions λ and θ are equivalent, and so the homogeneous spaces (G, G/H) and (G, M) can be identified.)

Remark 2

(G, M)

Any two isotropy subgroups of G are isomorphic: if x, y ∈ M such that g · x = y (for some element g ∈ G), then Gy = g Gx g −1 .

Remark 3

(G, M),

H = G0 ≤ G

The homogeneous space (G, M) is completely specified by (G, H). C.C. Remsing (Rhodes University)


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(7/12)

Definition

(G, H)

A Klein geometry is a pair (G, H), where G is a Lie group and H is a closed (Lie) subgroup such that (the coset manifold) G/H is connected. M = G/H is called the space of the Klein geometry G is called the group of motions (or the symmetry group) of the Klein geometry. (A Klein geometry (G, H) is called effective whenever the natural action of G on G/H is effective.)



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Homogeneous Spaces: A Few Ideas Definition(s)

(8/12) H = G0 ≤ G

(G, M),

The isotropy representation of H on (the vector space) To M: ι : H → GL (To M),

h 7→ (dh)1

The (induced) adjoint representation of H on (the vector space) g/h: Ad : H → GL (g/h),

X + h 7→ Adh X + h

Note The linear isotropy representation ι and the (induced) adjoint representation Ad of the (isotropy group H ≤ G) are equivalent. C.C. Remsing (Rhodes University)


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Homogeneous Spaces: A Few Ideas Equivalence (and local equivalence)

(9/12) (G, H)

(g, h)

Klein geometries (G, H) and (G, H) are called equivalent (or geometrically isomorphic) if there is a Lie group isomorphism ϕ : G → G such that ϕ(H) = H. Klein pairs (g, h) and (g, h) are called equivalent if there is a Lie algebra isomorphism φ : g → g such that φ(h) = h.

Equivalence implies local equivalence If the Klein geometries (G, H) and (G, H) are equivalent, then the corresponding Klein pairs (g, h) and (g, h), respectively, are equivalent.



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(10/12)

Local equivalence vs. equivalence

(Mostow, 1950)

Suppose that 1

the Klein geometries (G, H) and (G, H) are effective;

2

the isotropy groups H and H are connected;

3

the spaces M and M are (connected and) simply connected;

4

dim M = dim M ≤ 4.

If the Klein pairs (g, h) and (g, h) are equivalent, then the Klein geometries (G, H) and (G, H) are also equivalent.

Two-dimensional Klein geometries

(Komrakov et al., 1993)

There are exactly 23 types of two-dimensional (effective) Klein geometries, up to local equivalence (i.e., equivalence of the corresponding Klein pairs). C.C. Remsing (Rhodes University)


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Homogeneous Spaces: A Few Ideas Reductive homogeneous spaces

(11/12) H = Go ≤ G

(G, H),

A Klein geometry (G, H) is called reductive if there is an H-invariant subspace m, which is complementary to h: g = m ⊕ h and Adh m ⊂ m for all h ∈ H. (Hence the vector spaces m and To M are canonically isomorphic.)

Examples If H is compact, then (G, H) is reductive. If A := Rm o H ≤ Aff (n) is the affine extension of the linear Lie group H ≤ GL (m, R), then (A, H) is reductive. C.C. Remsing (Rhodes University)


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(12/12)

Result(s) If M = G/H admits a G-invariant Riemannian metric, then (G, H) is reductive. If (G, H) is reductive, then M = G/H admits a G-invariant affine connection. (Reductivity is not a necessary condition for the existence of an invariant connection.) There is a one-to-one correspondence between G-invariant tensor fields on M = G/H and H-invariant tensors on the vector space g/h. There is a one-to-one correspondence between G-invariant affine connections on M = G/H and linear maps α : g → End (g/h) s.t. α|h = ad : h → End (g/h)


and α(Adh X ) = Adh ◦ α(X ) ◦ Adh−1 .


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Two-Dimensional Klein Geometries

(1/8)

Euclidean geometry the Euclidean plane:

(Euc (2), O (2)) E2 = (M, σ);

QE : R2 → R,

M = R2 ,

σ = QE

x = (x1 , x2 ) 7→ x12 + x22

(QE is a positive definite quadratic form, which induces a metric dE ) the group of Euclidean motions: Euc (2) = R2 o O (2) = Isom (R2 , dE )

Note E2 = Euc (2)/O (2) is a Riemannian symmetric space. C.C. Remsing (Rhodes University)


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(2/8)

Spherical geometry the spherical plane:

(O (3), O (2)) M = S2 ⊂ R3 ,

(M, σ);

dS : S2 × S2 → R,

σ = dS

(x, y ) 7→ arccos (x • y ) ∈ [0, π]

(dS is an intrinsic metric) the group of spherical motions: G = O (3) = Isom (S2 , dS )

Note S2 = O (3)/O (2) is a Riemannian symmetric space. C.C. Remsing (Rhodes University)


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(3/8)

Elliptic geometry

(PO (3), O (2)) M = RP2 ,

the elliptic plane: E2`` = (M, σ); dP : RP2 × RP2 → R,

σ = dP

(±x, ±y ) 7→ min (dS (x, y ), dS (x, −y ))

(the spherical metric dS induces a metric dP on RP2 = S2 /∼ ) the group of elliptic motions: G = PO (3) = Isom (RP2 , dP )

(projective orthogonal group)

Note E2`` = PO (3)/O (2) is a Riemannian symmetric space. C.C. Remsing (Rhodes University)


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(4/8)

Hyperbolic geometry

(PO (1, 2), O (2))

the hyperbolic plane:

H2 = (M, σ);

M = H2 ⊂ R1,2 ,

H2 = x = (x1 , x2 , x3 ) : −x12 + x22 + x32 = −1, dH : H2 × H2 → R,

σ = dH

x1 > 0

(x, y ) 7→ arccosh (−x y )

the group of hyperbolic motions: G = PO (1, 2) = Isom (H2 , dH )

(projective pseudo-orthogonal group)

Note H2 = PO (1, 2)/O (2) is a Riemannian symmetric space. C.C. Remsing (Rhodes University)


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(5/8)

Galilean geometry

(Gal (2), R)

the Galilean plane:

G2 = (M, σ);

QI : R2 → R,

M = R2 ,

σ = QI

x = (x1 , x2 ) 7→ x12

(QI is a positive, degenerate quadratic form) the group of Galilean motions: (" Gal (2) = R2 o H,


H=


1 0 v 1

#

) : v ∈R

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Lorentzian geometry the Lorentzian plane:

(6/8)

(Lor (2), PO (1, 1)) L2 = (M, σ);

QL : R2 → R,

M = R2 ,

σ = QL

x = (x1 , x2 ) 7→ −x12 + x22

(QL is an indefinite, non-degenerate quadratic form) the group of Lorentzian motions: Lor (2) = R2 o H,


H = PO (1, 1)


(projective Lorentz group)

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(7/8)

Equi-affine geometry the equi-affine plane:

(ASL (2, R), SL (2, R)) A2 = (M, σ);

Φ : R2 × R2 → R2 , dV : R2 × R2 → R,

M = R2 ,

(x, y ) 7→ y − x (x, y ) 7→ det [x y ]

σ = (Φ, dV )

(structure map) (volume form)

the group of equi-affine motions: ASL (2, R) = R2 o H,



H = SL (2, R)

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(8/8)

Affine geometry the affine plane:

(Aff (2), GL (2, R)) A2 = (M, σ);

Φ : R2 × R2 → R2 ,

M = R2 ,

(x, y ) 7→ y − x

σ=Φ

(structure map)

the group of affine motions: Aff (2) = R2 o H,

H = GL (2, R)

Remark All these eight (classical) Klein geometries are reductive. (Other classical geometries–like Möbius or projective geometry–are non-reductive.) C.C. Remsing (Rhodes University)


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