1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Mathematicians and Spinors
Jean-Pierre BOURGUIGNON(CNRS-Institut des Hautes Études Scientifiques)

Inaugural Conference,
French-Japanese Laboratory of Mathematics and its Interactions

CNRS-Tokyo University
4 April, 2024

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Outline of the Lecture
Spinors were introduced in Geometry in 1913 by Élie CARTAN
and in Physics in 1928 by Paul-Adrien Maurice DIRAC;
When spinors have become fundamental objects in relation
with Quantum Mechanics in Physics, they are still objects
which are not familiar to a number of mathematicians;
This lecture presents their relevance in a number of questions
related to Differential Geometry and Global Analysis;
Since the Postface to the reedition of the book The Algebraic
Theory of Spinors by Claude CHEVALLEY I wrote in 1995,
arXiv lists some 2000 articles having “spinor” in their title;
Even the very restrictive expression “Spinorial Geometry”
appears 16 times in arXiv in the same period; on this topic, I
would like to support the following point of view: “the use ofspinors by mathematicians should lead to a Geometry on its
own; it should in particular shed light on numerous traditionalquestions of Riemannian Geometry”.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Outline of the Lecture (cont.)

Here is an outline of the topics to be covered:
1

The Strangeness of Spinors

2

Basic Facts on Spinors

3

Spin Manifolds and Spinorial Metrics

4

Natural Differential Operators on Spinor Fields

5

Harmonic Spinors

6

Spinors and Scalar Curvature

7

Killing Spinors

8...

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

1. The Strangeness of Spinors

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Spinors in Mathematics
Some general historical comments on how spinors appeared:
It is in 1913 that the concept of spinors appeared in the workof Élie CARTAN while he was doing a systematic classificationof the representations of orthogonal Lie algebras;
In 1937, he published an important book entitled La théorie
des spineurs; it is only in 1966 that this book was translated
into english;
The very last statement in the book says that, in a generalcoordinate system on a manifold, one cannot represent spinors
with finitely many components; what does that mean?
The reason for that is that spinors make sense only after ametric is fixed; this statement was misunderstood by many;
As we shall see later, it in connection with the Atiyah-Singer
Index Theorem that mathematicians started to give much
more attention to spinors, almost a rediscovery.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

How Spinors are Different
Some general comments on the relation of spinors with othergeometric objects:
When differential k-forms have a specific “dimension”, tospecify the “dimension” of a spinor is much trickier;
A precise formal comparison of spinors for different metrics
was not clear for a long time;
In 1933, this led Leopold INFELD, who wanted to use spinors
in the variational approach to the Theory of General Relativity,
to use spinors of a reference Minkowski metric when themetric used for the Physics was a general variable metric;
Isadore SINGER called my attention to wrong formulas usedin the physics literature until the early 1980s;
Paul GAUDUCHON and I solved the problem in 1992; in theend, the construction is quite simple but a little heavy;
For conformally related metrics, the comparison can be madeexplicit as the change of metric is scalar.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

How Spinors are Different (cont.)
The fact that spinors are difficult to comprehend continues to beperceptible to this day.
Here is a good example:
Lecturing in 2013 at the conference at IHÉS on the occasionof my leaving the directorship, Sir Michael ATIYAH entitled
his talk “What is a spinor?”;At the beginning of his presentation, he claimed: “I spent
most of my life working with spinors in one form or anotherand I do not know what a spinor is”’;
Therefore, it appeared natural to me to give the same title to
the Atiyah lecture I gave in January 2021, the first in theseries;Indeed, in this lecture, I discussed some of the peculiarfeatures of spinors that make them still difficult tocomprehend by mathematicians.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Further Comments on Spinors
Some further comments on spinors as geometric objects:
Roger PENROSE, in his monumental 2004 book The Road to
Reality, also asks the question: “What is a spinor?”
his answer is: “Essentially it is an object with turns into its
negative when it undergoes a complete rotation through 2π.
This may seem like an absurdity, because any classical objectof ordinary experience is always returned to its original state
under such a rotation, not to something else.”
he continues: “The very notion of a ’spinorial object’ issomewhat confusing and non-intuitive, and some people preferto resort to a purely (Clifford) algebraic approach to theirstudy. This certainly has its advantages... but I feel that it isimportant also not to lose sight of the geometry...”in an interview with Andrew HODGES in 2014, he said: “I
was mystified by spinors as they turn out to be square roots ofvectors, and I could not understand how you could do that.”1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

2. Spinors from an Algebraic Point of View

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Algebraic Properties of Spinors
To any n-dimensional Euclidean vector space (V , g ) is attached its
Clifford algebra
Cl(V , g ) = ⊗V /hx ⊗ x + g (x, x)1i .
Clifford algebras over C have a 2-fold periodicity (over R an 8-foldone):
if n = 2m, Cl(V , g ) is a simple algebra, hence
Cl(V , g ) = End(ΣV ) for a certain vector space ΣV ;if n = 2m + 1, then Cl(V , g ) = End(ΣV ) ⊕ End(Σ̃V ).
ΣV and Σ̃V are the spaces of spinors. They are 2m -dimensional.
These definitions make the dependence of spinors upon the metricquite implicit. What is explicitly defined is the projective space ofthe space of spinors.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Spinors and the Spin Group
Spinors are vectors in a fundamental representation space of thegroup Spinn , the universal cover of the group SOn for n ≥ 3.
The theory develops as follows:
A key ingredient in the whole theory is the exact sequence ofgroups0 −→ Z/2Z −→ Spinn −→ SOn −→ 1 ,which holds true for n ≥ 3.
The link with Clifford algebras goes as follows:
the group Spin(V , g ) can be realised as the multiplicative
subgroup of the Clifford algebra stabilising the image of V
inside Cl(V , g ) and satisfying a certain normalisation
condition;
therefore, through the adjoint representation, Spin(V , g ) acts
on V .

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Chirality
Of great importance in the study of spinors in even dimensions is
the notion of chirality.
Here what it is about:√under the action of ( −1)m vg , where vg is the volume
element determined by the metric g , the space ΣV splits intoΣV = Σ+ V ⊕ Σ− V ,the spaces of half-spinors or chiral spinors, both of dimension2m−1 ;
Clifford multiplication by v ∈ V exchanges chirality, hence
maps Σ+ V to Σ− V and vice-versa.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

3. Spin Manifolds and Spinorial Metrics

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Spin Manifolds
The purpose is of course to carry these constructions over to anoriented Riemannian manifold (M, g ).
Here is how it goes:
For tensor fields, the usual constructions lead to a bundle
obtained from the tangent and cotangent bundles TM andT ∗ M over M by the appropriate tensor bundle operations;
Another equivalent approach starts from the SOn -principal
bundle of g -orthonormal oriented bases on M, denoted by
SOg M −→ M;
Any tensor bundle can be recovered from SOg M −→ M by
the associated bundle construction.
For a given tensor representation ρ : SOn −→ RN , the associatedtensor bundle is SOg M ×ρ RN −→ M, where ×ρ means taking thequotient of the product space by the equivalence relation(b, ξ) ≡ (b ◦ u, ρ(u)−1 (ξ)) for u ∈ SOn .

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Spinor Bundles and Bundles of Spinors
Over the oriented Riemannian manifold (M, g ), one needs todispose of a Spinn -principal bundle Spinγ M covering SOg M, to beviewed as the bundle of spinorial bases for the spinorial metric γ.
The construction goes as follows:
The possibility of finding such a Spinγ M is related to thecohomology sequence attached to the sequence of groups:
0 −→ Z/2Z −→ Spinn −→ SOn −→ 1 ;A necessary and sufficient condition for the existence of a spin
structure needed to give rise to Spinγ M is that the second
Stiefel-Whitney class w2 (M) ∈ H 2 (M, Z/2Z) vanishes; onethen says that M is a spin manifold;
The ambiguity lies in H 1 (M, Z/2Z);
The bundle of spinors associated to the spinorial metric γ isthe associated bundle Σγ M = Spinγ M ×Spinn Σn ;The bundle in Clifford algebras Clg (M) −→ M acts on the
spinor bundle via pointwise Clifford multiplication.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

4. Natural Differential Operators on Spinor Fields

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Covariant Derivative of Spinor Fields
The principal bundle Spinγ M −→ M inherits a natural connection
form from the Riemannian connection form on the Riemannian
frame bundle SOg M −→ M, hence the spinor bundle inherits acovariant derivative by the associated bundle construction.
Here are the consequences:
Let us recall that, on the space Γ(E ) of sections of a vector
bundle E −→ M, covariant derivatives ∇ are first orderdifferential operators mapping Γ(E ) to Γ(T ∗ M ⊗ E ) with theidentity as principal symbol σ(∇) : T ∗ M ⊗ E −→ T ∗ M ⊗ E ;This comes from the axiom of covariant derivatives that
requires that, for f ∈ C ∞ (M) and s ∈ Γ(E ),∇(f s) = df ⊗ s + f ∇s ;For a spinor field ψ, we will denote its covariant derivative by
D γ ψ if we want to be emphatic about the dependence of thecovariant derivative on the spinorial metric γ.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Natural Differential Operators on Spinor Fields
On the space of spinor fields only two first order differential
operators are universally defined, and combinations thereof:
This is because T ∗ M ⊗ Σγ M decomposes into two invariant
3/2
subspaces, a copy of Σγ M and another space Σγ M;
The projection onto the first factor gives rise to the Dirac
operator D that maps spinor fields to spinor fields, and isdefined, for a spinor field ψ, bynX
Dγ ψ =
ei .Deγi ψ ,i=1

where (ei ) denotes an orthonormal basis of the tangent space;
The projection onto the second factor gives rise to thePenrose operator P, which maps Γ(Σγ M) to Γ(T ∗ M ⊗ Σγ M),
and is defined for X ∈ TM and ψ is a spinor field by
1(P γ ψ)(X ) = DXγ ψ + X .Dγ ψ .
n

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Dirac Operators
Here are some important properties of the Dirac operator:
In a Riemannian setting, Dγ is a square root of the
Laplace-Beltrami operator, hence an elliptic operator;
Its principal symbol in the cotangent direction ξ is given by
Clifford multiplication, i.e. σξ (ψ) = ]γ (ξ).ψ ;The operator Dγ is self-adjoint;
In even dimensions, Dγ exchanges the chirality of spinors,
hence non-trivial eigenspinors for Dγ have necessarily
components of both chiralities, unless they are harmonic;
For a spinor field ψ, the Schrödinger-Lichnerowicz formulareads
1(Dγ )2 ψ = D γ ∗ D γ ψ + Scalg ψ ,4∗γ
where D denotes the adjoint of the covariant derivative D γ
and Scalg the scalar curvature of g .

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

5. Harmonic Spinors

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Harmonic Spinors
Definition
For a spinorial metric γ, a harmonic spinor ψ is a spinor field ψlying in the kernel of the Dirac operator Dγ , i.e. Dγ ψ = 0.
On an n-dimensional spinorial manifold (M, γ), we denote thespace of harmonic spinors by H(M, γ).
If n = 2m, for Dγ + (resp. Dγ − ) the restriction of the Dirac
operator to positive (resp. negative) spinor fields, we consider
positive (resp. negative) harmonic spinors H± (M, γ) = ker Dγ ± .
Note that, when n = 8` + 4, the spinor representation is naturally
quaternionic, hence H+ (M, γ) and H− (M, γ) are automaticallyeven-dimensional complex vector spaces.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Dirac Operators and Index Theorems
When applied to the Dirac operator Dγ + acting in even dimensions
on positive spinor fields, the Atiyah-Singer Index Theorem leads tothe following crucial statement:
Theorem (M.F. ATIYAH, I.M. SINGER)
On a compact 4k-dimensional spin manifold M endowed with aspinorial metric γ,
Index Dγ + = dim H+ (M, γ) − dim H− (M, γ) = Â(M) .
This statement has a number of consequences on the relation
between Metric Geometry and Topology:
It is the basis for important integrality theorems;
It gives a topological lower bound for dim H;
Already in 1962, André LICHNEROWICZ identified theÂ-genus as obstructing the existence of a positive scalar
curvature metric on a compact spin manifold in dimension 4k.
.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

6. Spinors and Scalar Curvature1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Scalar Curvature
For surfaces the curvature reduces to the Gaussian curvature. In
dimension n ≥ 3, the scalar curvature function is the weakest
curvature invariant of a Riemannian manifold.
Several important results have been obtained concerning the scalar
curvature in the 1970s:
In 1975, Jerry L. KAZDAN and Frank WARNER proved that,for n ≥ 3, any function that is negative somewhere is thescalar curvature of a metric on an n-dimensional manifold;
The analogous statement for metrics with positive scalar
curvature is not true for some manifolds as shown by the resultof André LICHNEROWICZ combining the Atiyah-Singer Index
Theorem and the formula for the square of the Dirac operator;
In 1973, Robert GEROCH posed a challenging question: does
the n-torus admit a metric with positive scalar curvature?

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Positive Scalar Curvature
Determining whether a given manifold admits a positive scalar
curvature metric has generated many works in the last 60 years.
We concentrate on compact manifolds without boundary:In 1974, using more sophisticated KO-invariants, Nigel
HITCHIN proved that their non-vanishing obstructs positive
scalar curvature metrics; this covers some exotic spheres;
In 1980, Misha GROMOV and H. Blaine LAWSON showed
that, for simply connected n-manifolds (n ≥ 5), the answer
depends only on the Spin-bordism class of the manifold;In the same article, they prove that all simply connected
non-spin n-manifolds admit positive scalar curvature metrics
for n ≥ 5 ;In 1992, a critical step was achieved by Stephan STOLZ whoproved that, for simply connected spin manifolds, theLichnerowicz-Hitchin obstructions form a complete set.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Positive Scalar Curvature (cont.)
The non-simply connected case has mobilised a lot of attention.
Here are critical results accumulated over the years:
In 1979 and later, Richard SCHOEN and YAU Shing Tungproved a so-called Splitting Theorem establishing theinheritance of positivity of the scalar curvature on any
non-trivial integral homology class a priori restricted todimensions n ≤ 7; but they could cover the case of the torus;
In 1980, Misha GROMOV and Blaine LAWSON constructed
new obstructed manifolds: the so-called enlargeable manifolds;
Later, many refinements in presence of a fundamental groupwere obtained in particular by Jonathan ROSENBERG andStephan STOLZ involving the algebraic structure of the group;
In 2012, there was some beautiful work in the same vein byZizhou TANG, Yuquan XIE and Wenjiao YAN;
In 2022, Stephan STOLZ produced a comprehensive report on
the problem.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

7. Killing Spinors

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Killing Spinors
Definition (R. PENROSE)
A Killing spinor ψ is a spinor field lying in the kernel of P and an
eigenspinor for D. Its characteristic equation is, for some λ ∈ C,
1
∀X ∈ TM, DX ψ + λ X .ψ = 0 .
nSome facts about Killing spinors:
The 1-form ξψ defined on X ∈ TM by ξψ (X ) = (X .ψ, ψ) isdual to a Killing vector field, i.e. an infinitesimal isometry;
Other components of ψ ⊗ ψ̄ also satisfy interesting conditions;
A definition of a supersymmetric transformation can go as
follows : one maps fermionic fields (such as spinor fields) ϕ tobosonic fields (such as 1-forms) by ϕ 7→ (X 7→ <e(X .ϕ, ψ)) ;The curvature tensor of D γ acting on ψ is very special,namely, for all X , Y ∈ TM, RX ,Y ψ = n12 λ2 (X .Y − Y .X ).ψ ;It follows that Ricg = 4 n−1
λ2 g .
n2

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Killing Spinors (cont.)
It is useful to discuss cases according to the eigenvalue λ :If λ = 0, then the Killing spinor is parallel, and hence themetric has reduced holonomy;
If λ ∈ i R∗ , then, if g is complete, M must be non compact;
If λ ∈ R∗ , then the Ricci curvature is uniformly positive, andby Myers’ Theorem, M is compact if the metric g is complete.
The key construction for Killing spinors, is due to Christian BÄR.
It goes as follows:
Construct the cone CM = R+∗ × M over M and endow itwith the cone metric ḡ = dr 2 + r 2 g ;Then, through an identification of an action of the group
Spinn+1 within the Clifford algebra Clg (M), map spinor fields
on M into spinor fields on CM ;Through this identification, Killing spinors on M are mappedto parallel fields on CM .

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Manifolds Admitting Killing Spinors
Many examples are connected to holonomy considerations:
Manifolds with parallel spinors have necessarily a reduced
holonomy; a classification has been provided by Nigel
HITCHIN: the possibilities are SUm Spin7 and G2 ;Manifolds with imaginary Killing spinors (actually the caseforces the manifold to be non compact if complete) have beenclassified by Helga BAUM: they are hyperbolic spaces or
special warped products of R with a manifold with a parallelspinor;The classification for real Killing spinors is a bit more involved;
For 2 ≤ n , the standard sphere has only one spin structure;
for the standard metric, the bundle of spinors is trivialised byKilling spinors that are induced by constant spinors in Rn+1 ;In even dimensions other than 6, only standard spheres carry
Killing spinors; in dimension 6, one also finds the manifolds
endowed with a nearly Kähler non-Kähler metric.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Manifolds Admitting Killing Spinors (cont.)
Other examples are interesting from a geometric or even a physicalpoint of view:
In dimension 5, Thomas FRIEDRICH and Ines KATH provedthat one finds only manifolds with an Einstein-Sasaki metric;
this was generalised by Christian BÄR in all dimensions 4q + 1;
In dimensions 4q + 3 (for 2 ≤ q), Christian BÄR showed that
one has to add the Sasaki 3-manifolds, a case studied further
by Andrei MOROIANU;
In the remaining dimension 7, the extra family to add
corresponds to manifolds for which the cone built over them
carries a metric with Spin7 holonomy;
The so-called squashed 7-sphere is a very interesting metric on
the sphere as one can come to it from the purely Riemannian
point of view (an exotic Einstein metric on the sphere besides
the standard one) or from a supergravity point of view.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Generalised Killing Spinors
The notion of Killing spinors has been generalised in different ways.
One has to do with sub-manifold theory.
Here is a construction due to Christian BÄR, Paul GAUDUCHON
and Andrei MOROIANU:
On a Riemannian manifold (M, g ) consider a field of
symmetric linear maps S satisfying the Codazzi equation on
M and assume that there exists a generalised Killing Spinor ψsatisfying, for any X ∈ TM, the equation
1
DX ψ − S(X ).ψ = 0 ;2
Let I ⊂ R be an interval parametrised by t, and endow I × M
with the metric dt 2 + gt , where gt = g ◦ (Id − tS)2 ;Then, there is a parallel spinor field on I × M whose
restriction to {0} × M is precisely ψ ;Any generalised Killing spinor on S 8 is standard;Works on generalised Killing spinors continue to this day.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

8. ...

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

Some Further Comments
Spinors continue to both attract the attention of mathematicians
but also challenge their geometrical feelings.
Admitting a spin structure is rather restrictive, excluding forexample even-dimensional complex projective spaces.
One way to get around this has been to introduce
Spinc -structures, i.e. to consider an extension of Spinn by U1 ;There is still a topological condition for the existence of sucha structure but it is satisfied by all complex manifolds;
A Dirac operator can be defined in this context; it is verymuch connected to the Dolbeault operator ∂¯ + ∂¯∗ ;It found a number of applications in Theoretical Physics, in
relation with the theory of anomalies.
Recently, were considered metrics on Riemann surfaces withextremal eigenvalues of the Dirac operator: Mikhail KARPUKHIN,
Antoine MÉTRAS and Iosif POLTEROVICH show that theycorrespond to harmonic maps into complex projective spaces.

1. Strangeness 2. Basics 3. Spin Manifolds 4. Natural Operators 5. Harmonic Spinors 6. Scalar Curv. 7. Killing Spinors 8...

I thank you for your attention.

Jean-Pierre BOURGUIGNON
Institut des Hautes Études Scientifiques
35, route de Chartres
F-91440 BURES-SUR-YVETTE(France)
JPB@ihes.fr