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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \newcommand{\exercise}{% {\bf \green EXERCISE:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf K\"ahler geometry \\[15mm] \small lecture 2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf University of Science and Technology China, Hefei \\[2mm] July 09, 2012 } \end{center} \newpage {\bf \blue Some textbooks} Arthur L. Besse, Einstein Manifolds, Springer-Verlag (Berlin, 1987). P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York (1978). D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, 2000 Jean-Pierre Demailly, Complex analytic and differential geometry, \\ \url {http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf} Claire Voisin, Hodge theory and complex algebraic geometry, Cambridge University Press, 2007 D. Huybrechts, Complex geometry: an introduction, Springer Universitext, 2004 \newpage {\bf \blue REMINDER: Complex structure on vector spaces} \definition Let $V$ be a vector space over $\R$, and $I:\; V \arrow V$ an automorphism which satisfies $I^2 = - \Id_V$. Such an automorphism is called {\bf\blue a complex structure operator} on $V$. \definition The vector space over $\C$ with the same basis is called {\bf \blue a complexification} of $V$, denoted $V \otimes_\R \C$. \claim For an appropriate basis in $V \otimes_\R \C$, the complex structure operatorcan be written as {\scriptsize\[ I=\begin{pmatrix} \1 &0&0 &\hdotsfor{1} &0&0&0\\ 0&\1 &0 &\hdotsfor{1} &0&0&0\\ 0&0&\1 &\hdotsfor{1} &0&0&0\\ \vdots&\vdots&\vdots& \ddots &\vdots&\vdots&\vdots\\ 0&0&0 &\hdotsfor{1} &-\1&0&0\\ 0&0&0 &\hdotsfor{1} &0&-\1&0\\ 0&0&0 &\hdotsfor{1} &0&0&-\1 \end{pmatrix}, \]} with the eigenspaces of equal dimension. \newpage {\bf \blue REMINDER: Hermitian structures} \definition Let $(V,I)$ be a real vector space with a complex structure. A scalar product is called {\bf \blue $I$-invariant}, if $g(Ix,Iy)=g(x,y)$. An $I$-invariant positive definite scalar product on $(V,I)$ is called {\bf \blue an Hermitian metric on $V$}, and $(V,I,g)$ -- an Hermitian space. \remark Let $I$ be a complex structure operator on a real vector space $V$, and $g$ -- a Hermitian metric. Then {\bf \red the bilinear form $\omega(x,y) := g(x, Iy)$ is skew-symmetric.} Indeed, $\omega(x,y) = g(x, Iy) = g(Ix, I^2y) = -g(Ix, y) = -\omega(y, x)$. \definition A skew-symmetric form $\omega(x,y)$ is called {\bf \blue an Hermitian form on $(V,I)$}. \remark In the triple $I, g, \omega$, each element can recovered from the other two. \newpage {\bf \blue REMINDER: tensor product} \definition Let $V, W$ be vector spaces over a field $k=\R$ or $\C$, and $V^*, W^*$ the dual spaces. Denote by $\Bil(V\times W, k)$ the space of bilinear maps from $V,W$ to $k$, and let $V \otimes W$ denote $\Bil(V^*\times W^*, k)$, where $V^*,W^*$ are dual spaces to $V, W$. The space $V \otimes W$ is called {\bf \blue the tensor product} of $V, W$. \definition Given $v, w\in V, W$, one has an element $v\otimes w \in \Bil(V^*\times W^*, k)$, mapping a pair of functionals $\lambda\in V^*, \mu \in W^*$ to $\lambda(v)\mu(w)$. This vector is called {\bf \blue the tensor product} of $v$ and $w$. \claim If $\{v_i, i=1,..., n\}$ is a basis in $V$, $\{w_j, j = 1, ..., m\}$ a basis in $W$, {\bf \purple the vectors $\{v_i \otimes w_j, i=1,..., n, j = 1, ..., m\}$ give a basis in $V \otimes W$.} In particuler, {\bf \red $V \otimes W$ is $(\dim V \cdot \dim W)$-dimensional.} \remark The tensor product is uniquely defined by the following {\bf \blue universal property}. Each bilinear map $B:\; (V, W) \arrow k$ can be extended uniquely to the map $B^\otimes:\; V \otimes W \arrow k$, in such a way that $B^\otimes(v \otimes w) = B(v,w)$. \exercise {\bf \purple Prove this}. \newpage {\bf \blue REMINDER: tensor algebra} \definition Given several vector spaces $V_1, ..., V_n$, their {\bf \blue tensor product} $V_1\otimes V_2\otimes ... \otimes V_n$ is defined as $V_1\otimes (V_2 \otimes (V_2 \otimes ... V_n))...)$. \claim The tensor product operation {\bf \red is commutative and associative.} Moreover, {\bf \purple the space $V_1\otimes V_2\otimes ... \otimes V_n$ is isomorphic to the space $B(V_1^*, V_2^*, ..., V_n^*)$} of polylinear maps from $V_1^*, V_2^*, ..., V_n^*$ to $k$. \exercise {\bf \purple Prove this}. \remark Given vectors in $v\in V_1\otimes V_2\otimes ... \otimes V_n$ and $w\in W_1\otimes W_2\otimes ... \otimes W_m$, the tensor product $v \otimes w$ sits in $V_1\otimes V_2\otimes ... \otimes V_n\otimes W_1\otimes W_2\otimes ... \otimes W_m$. \claim This defines the structure of an algebra on $T^\otimes V= k \oplus V \oplus V\otimes V \oplus ... \oplus V^{\otimes n}$, where $V^{\otimes n}$ is a tensor product of $n$ copies of $V$. \exercise {\bf \purple Prove this}. \definition The algebra $T^\otimes V$ is called {\bf \blue the tensor algebra}, or {\bf \blue free algebra} generated by $V$. \newpage {\bf \blue REMINDER: The Grassmann algebra} \exercise Prove that {\bf \purple any algebra generated by $V$ can be obtained as a quotient of $T^\otimes V$ by an ideal.} \exercise Let $v_1, ..., v_n\in V$ be a basis. Prove that {\bf \purple the polynomial algebra $k[v_1, ..., v_n]$ is a quotient of $T^\otimes V$ by an ideal generated by $x\otimes y - y\otimes x$, for all $x,y \in V$.} \definition {\bf \blue A Grassmann algebra} $\Lambda^* V$ is a quotient of $T^\otimes V$ by an ideal generated by $x\otimes y + y\otimes x$, for all $x,y \in V$. The multiplication in $\Lambda^* V$ is denoted by $x, y \arrow x\wedge y$, called {\bf \blue the wedge product}. {\bf \green Properties of Grassmann algebra:} 1. $\dim \Lambda^i V:= \binom{\dim V}{i}$, $\dim \Lambda^* V=2^{\dim V}$. 2. $\Lambda^*(V \oplus W)= \Lambda^*(V) \otimes \Lambda^*(W)$. \newpage {\bf \blue REMINDER: Vector fields} \definition Let $X$ be the vector field on a manifold $M$, and $f$ a function. Denote by $\Lie_X f$ {\bf \blue the derivatiive} of $f$ along $X$. \definition {\bf \blue A derivation} on a commutative ring is a map $R \stackrel d \arrow R$ satisfying {\bf \blue the Leibniz identity} $d(xy) = d(x) y + x d(y)$. \theorem Each derivation of the ring $C^\infty M$ of smooth functions on $M$ is given by a vector field $X$; {\bf \red this correspondence is bijective.} \remark This can be used as a definition of a vector field. \exercise Prove that {\bf \red a commutator of two derivations is again a derivation.} \remark Vector fields are the same as derivations of $C^\infty M$. This allows us to define {\bf \blue the commutator of two vector fields} as the commutator of the corresponding derivations. \definition Denote by $TM$ the bundle of vector fields, and by $\Lambda^1 M$ or $T^*$ the dual bundle, called {\bf \blue the bundle of 1-forms}. For any $f\in C^\infty M$, the operation $X \arrow \Lie_X f$ is linear as a function of $X$, hence it defines a section of $T^*M$. We denote this section $df$, and call it {\bf \blue the differential} of $f$. \newpage {\bf \blue REMINDER: de Rham algebra} \definition Let $\Lambda^* M$ denote the vector bundle with the fiber $\Lambda^*T^*_xM$ at $x\in M$ ($\Lambda^*T^*M$ is the Grassman algebra of the cotangent space $T^*_x M$). The sections of $\Lambda^i M$ are called {\bf \blue differential $i$-forms}. The algebraic operation ``wedge product'' defined on differential forms is $C^\infty M$-linear; the space $\Lambda^* M$ of all differential forms is called {\bf \blue the de Rham algebra}. \remark $\Lambda^0 M = C^\infty M$. \theorem {\bf \red There exists a unique operator $C^\infty M\stackrel d \arrow \Lambda^1 M\stackrel d \arrow \Lambda^2 M \stackrel d \arrow \Lambda^3 M \stackrel d \arrow ...$ satisfying the following properties} 1. On functions, $d$ is equal to the differential.\\ 2. $d^2=0$ \\ 3. $d(\eta \wedge \xi) = d(\eta) \wedge \xi + (-1)^{\tilde \eta}\eta \wedge d(\xi)$, where $\tilde \eta=0$ where $\eta\in \lambda^{2i}M$ is {\bf \blue an even form,} and $\eta\in \lambda^{2i+1}M$ is {\bf \blue odd.} \definition The operator $d$ is called {\bf \blue de Rham differential}. \exercise {\bf \purple Prove it}. \definition A form $\eta$ is called {\bf \blue closed} if $d\eta=0$, {\bf \blue exact} if $\eta in \im d$. The group $\frac{\ker d}{\im d}$ is called {\bf \blue de Rham cohomology} of $M$. \newpage {\bf \blue REMINDER: Complex manifolds} {\bf\green DEFINITION:} Let $M$ be a smooth manifold. An {\bf \blue almost complex structure} is an operator $I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$. {\bf \purple The eigenvalues of this operator are $\pm \1$.} The corresponding eigenvalue decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$. {\bf\green DEFINITION:} An almost complex structure is {\bf \blue integrable} if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$. In this case $I$ is called {\bf \blue a complex structure operator}. A manifold with an integrable almost complex structure is called {\bf \blue a complex manifold}. {\bf\green THEOREM:} (Newlander-Nirenberg)\\ {\bf \red This definition is equivalent to the usual one.} \remark The commutator defines a $\C^\infty M$-linear map\\ $N:=\Lambda^2(T^{1,0})\arrow T^{0,1}M$, called {\bf \blue the Nijenhuis tensor} of $I$. {\bf \purple One can represent $N$ as a section of $\Lambda^{2,0}(M) \otimes T^{0,1}M$.} \example {\bf \red Symmetric spaces.} \example {\bf \red $\C P^n$.} \newpage {\bf \blue K\"ahler manifolds} {\bf\green DEFINITION:} An Riemannian metric $g$ on an almost complex manifiold $M$ is called {\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$. In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$, hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric. {\bf\green DEFINITION:} The differential form $\omega\in \Lambda^2(M)$ is called {\bf \blue the Hermitian form} of $(M,I,g)$. \remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}. {\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$ is called {\bf \blue K\"ahler} if $d\omega=0$. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$, and $\omega$ {\bf \blue the K\"ahler form}. {\bf \green Definition:} Let $M=\C P^n$ be a complex projective space, and $g$ a $U(n+1)$-invariant Riemannian form. It is called {\bf \blue Fubini-Study form on $\C P^n$}. The Fubini-Study form is obtained by taking arbitrary Riemannian form and averaging with $U(n+1)$ using the Haar measure on $U(n+1)$. \exercise Prove that {\bf \red the Fubini-Study form is unique} (up to a constant multiplier). \newpage {\bf\blue Examples of K\"ahler manifolds.} {\bf \green Remark:} For any $x\in \C P^n$, the stabilizer $St(x)$ is isomorphic to $U(n)$. Fubini-Study form on $T_x\C P^n= \C^n$ is $U(n)$-invariant, hence unique up to a constant. {\bf \green Claim:} {\bf \red Fubini-Study form is K\"ahler.} Indeed, $d\omega\restrict x$ is a $U(n)$-invariant 3-form on $\C^n$, but such a form must vanish, because $-\Id\in U(n)$ \remark {\bf \purple The same argument works for all symmetric spaces.} \definition {\bf \blue An almost complex submanifold} $X \subset M$ of an almost complex manifold $(M,I)$ is a smooth submanifold which satisfies $I(TX)\subset TX$. \exercise Let $X \subset M$ be an almost complex submanifold of $(M,I)$, where $I$ is integrable. {\bf \purple Prove that $(X, I\restrict{TX})$ is a complex manifold.} \definition In this situation, $X$ is called {\bf \blue a complex submanifold} of $M$. {\bf \green Corollary:} {\bf \red Every projective manifold (complex submanifold of $\C P^n$) is K\"ahler.} Indeed, a restriction of a closed form is again closed. \newpage {\bf \blue Connections} {\bf \green Notation:} Let $M$ be a smooth manifold, $TM$ its tangent bundle, $\Lambda^iM$ the bundle of differential $i$-forms, $C^\infty M$ the smooth functions. {\bf \purple The space of sections of a bundle $B$ is denoted by $B$.} \definition A {\bf\blue connection} on a vector bundle $B$ is a map $B \stackrel \nabla \arrow \Lambda^1 M \otimes B$ which satisfies \[ \nabla(fb) = df \otimes b + f \nabla b\] for all $b\in B$, $f\in C^\infty M$. \remark A connection $\nabla$ on $B$ gives a connection $B^* \stackrel {\nabla^*} \arrow \Lambda^1 M \otimes B^*$ on the dual bundle, by the formula \[ d(\langle b, \beta\rangle) = \langle \nabla b, \beta\rangle+ \langle b, \nabla^*\beta\rangle \] These connections are usually denoted {\bf \red by the same letter $\nabla$.} \remark For any tensor bundle ${\cal B}_1:= B^*\otimes B^* \otimes ... \otimes B^* \otimes B\otimes B \otimes ... \otimes B$ {\bf \green a connection on $B$ defines a connection on ${\cal B}_1$} using the Leibniz formula: \[ \nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2). \] \newpage {\bf \blue Torsion} \definition {\bf \blue The torsion} of a connection $\Lambda^1 \stackrel \nabla \arrow \Lambda^1 M \otimes \Lambda^1M$ is a map $\Alt \circ \nabla - d$, where $\Alt:\; \Lambda^1 M \otimes \Lambda^1M\arrow \Lambda^2 M$ is exterior multiplication. It is a map $T_\nabla:\; \Lambda^1M \arrow \Lambda^2 M$. \exercise {\bf \red Prove that torsion is a $C^\infty M$-linear.} \remark The dual operator $x, y \arrow \nabla_x Y - \nabla_y X-[X,Y]$ is also called {\bf \blue the torsion of $\nabla$}. It is a map $\Lambda^2 TM \arrow TM$. \exercise {\bf \purple Prove that these two tensors are dual.} \definition Let $(M, g)$ be a Riemannian manifold. A connection $\nabla$ is called {\bf \blue orthogonal} if $\nabla(g) =0$. It is called {\bf \blue Levi-Civita} if it is torsion-free. \theorem (``the main theorem of differential geometry'')\\ {\bf \red For any Riemannian manifold, the Levi-Civita connection exists,\\ and it is unique}. \newpage {\bf \blue Levi-Civita connection and K\"ahler geometry} {\bf \green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian manifold. {\bf \purple Then the following conditions are equivalent.} (i) {\bf \red The complex structure $I$ is integrable, and the Hermitian form $\omega$ is closed.} (ii) One has {\red $\nabla(I)=0$,} where $\nabla$ is the Levi-Civita connection. \remark {\bf \purple The implication (ii) $\Rightarrow$ (i) is clear.} Indeed, $[X,Y]=\nabla_X Y - \nabla_Y X$, hence it is a $(1,0)$-vector field when $X, Y$ are of type (1,0), and then {\bf \purple $I$ is integrable}. Also, {\bf \purple $d\omega=0$, because $\nabla$ is torsion-free,} and $d\omega= \Alt(\nabla\omega)$. The implication (i) $\Rightarrow$ (ii) is proven by the same argument as used to construct the Levi-Civita connection. \newpage {\bf \blue Holonomy group} \definition (Cartan, 1923) Let $(B,\nabla)$ be a vector bundle with connection over $M$. For each loop $\gamma$ based in $x\in M$, let $V_{\gamma, \nabla}:\; B\restrict x \arrow B\restrict x$ be the corresponding parallel transport along the connection. The {\bf \blue holonomy group} of $(B,\nabla)$ is a group generated by $V_{\gamma, \nabla}$, for all loops $\gamma$. If one takes all contractible loops instead, $V_{\gamma, \nabla}$ generates {\bf \blue the local holonomy}, or {\bf \blue the restricted holonomy} group. \remark A bundle is {\bf \blue flat} (has vanishing curvature) {\bf\purple if and only if its restricted holonomy vanishes.} \remark If $\nabla(\phi)=0$ for some tensor $\phi\in B^{\otimes i}\otimes (B^*)^{\otimes j}$, {\bf \red the holonomy group preserves $\phi$.} \definition {\bf \blue Holonomy of a Riemannian manifold} is holonomy of its Levi-Civita connection. \example Holonomy of a Riemannian manifold lies in $O(T_x M, g\restrict x)=O(n)$. \example Holonomy of a K\"ahler manifold lies in $U(T_x M, g\restrict x, I \restrict x)=U(n)$. \remark The holonomy group {\bf \red does not depend on the choice of a point $x\in M$.} \newpage {\bf \blue Curvature of a connection} Let $M$ be a manifold, $B$ a bundle, $\Lambda^i M$ the differential forms, and $\nabla:\; B \arrow B \otimes \Lambda^1M$ a connection. We extend $\nabla$ to $B \otimes \Lambda^iM\stackrel\nabla \arrow B \otimes \Lambda^{i+1}M$ in a natural way, using the formula \[ \nabla(b \otimes \eta) = \nabla(b)\wedge \eta + b \otimes d\eta, \] and define {\bf \blue the curvature $\Theta_\nabla$} of $\nabla$ as $\nabla\circ \nabla:\; B \arrow B\otimes \Lambda^2M$. \claim {\bf \red This operator is $C^\infty M$-linear.} \remark We shall consider $\Theta_\nabla$ as an element of $\Lambda^2M \otimes \End B$, that is, an $\End B$-valued 2-form. \remark Given vector fields $X, Y\in TM$, the curvature can be written in terms of a connection as follows \[ \Theta_\nabla(b)= \nabla_X\nabla_Yb - \nabla_Y\nabla_X B - \nabla_{[X,Y]}b. \] \claim Suppose that the structure group of $B$ is reduced to its subgroup $G$, and let $\nabla$ be a connection which preserves this reduction. This is the same as to say that the connection form takes values in $\Lambda^1 \otimes \goth g(B)$. {\bf \purple Then $\Theta_\nabla$ lies in $\Lambda^2M \otimes \goth g(B)$.} \newpage {\bf \blue The Lasso lemma} \definition A {\bf \blue lasso} is a loop of the following form: {\begin{center} \epsfig{file=Lasso.png,width=0.35\linewidth} \end{center}} The round part is called {\bf \blue a working part} of a loop. \remark {\bf \blue (``The Lasso Lemma'')} Let $\{U_i\}$ be a covering of a manifold, and $\gamma$ a loop. Then {\bf \purple any contractible loop $\gamma$ is a product of several lasso, with working part of each inside some $U_i$.} \newpage {\bf \blue The Ambrose-Singer theorem} \definition Let $(B, \nabla)$ be a bundle with connection, $\Theta\in \Lambda^2(M)\otimes \End(B)$ its curvature, and $a,b\in T_x M$ tangent vectors. An endomorphism $\Theta(a,b)\in \End(B)\restrict x$ is called {\bf \blue a curvature element}. \theorem {\bf \blue (Ambrose-Singer)} The restricted holonomy group of $B, \nabla$ at $z\in M$ is a Lie group, {\bf \red with its Lie algebra generated by all curvature elements $\Theta(a,b)\in \End(B)\restrict x$ transported to $z$ along all paths.} \remark Its proof follows from the Lasso lemma. \newpage {\bf \blue Holonomy representation} \definition Let $(M,g)$ be a Riemannian manifold, $G$ its holonomy group. A {\bf\blue holonomy representation} is the natural action of $G$ on $TM$. \theorem (de Rham) Suppose that the holonomy representation is not irreducible: $T_xM=V_1 \oplus V_2$. Then $M$ locally splits as $M=M_1 \times M_2$, with $V_1= TM_1$, $V_2=TM_2$. {\bf\green Proof. Step 1:} Using the parallel transform, we extend $V_1 \oplus V_2$ to a {\bf \purple splitting of vector bundles $TM = B_1 \oplus B_2$, preserved by holonomy.} {\bf\green Step 2:} The sub-bundles $B_1$, $B_2 \subset TM$ {\bf \purple are integrable:} $[B_1, B_1] \subset B_i$ (the Levi-Civita connection is torsion-free) {\bf\green Step 3:} Taking the leaves of these integrable distributions, {\bf \purple we obtain a local decomposition $M=M_1 \times M_2$, with $V_1= TM_1$, $V_2=TM_2$. } {\bf\green Step 4:} Since the splitting $TM = B_1 \oplus B_2$ is preserved by the connection, {\bf \purple the leaves $M_1, M_2$ are totally geodesic.} {\bf\green Step 5:} Therefore, {\bf \red locally $M$ splits (as a Riemannian manifold)}: \\ $M=M_1\times M_2$, where $M_1, M_2$ are any leaves of these foliations. \endproof \newpage {\bf \blue The de Rham splitting theorem} \corollary Let $M$ be a Riemannian manifold, and $\Hol_0(M)\stackrel \rho \arrow \End(T_xM)$ a reduced holonomy representation. Suppose that $\rho$ is reducible: $T_xM = V_1\oplus V_2 \oplus ...\oplus V_k$. {\bf \red Then $G=\Hol_0(M)$ also splits: $G= G_1\times G_2 \times ...\times G_k$,} with each $G_i$ acting trivially on all $V_j$ with $j\neq i$. {\bf \green Proof:} Locally, this statement follows from the local splitting of $M$ proven above. To obtain it globally in $M$, use the Lasso Lemma. \endproof \theorem (de Rham) A complete, simply connected Riemannian manifold with non-irreducible holonomy {\bf \red splits as a Riemannian product.} \remark It is easy to find non-complete or non-simply connected counterexamples to de Rham theorem. \theorem (Simons, 1962) Let $M$ be a manifold with irreducible holonomy. {\bf \red Then either $M$ is locally symmetric, or $\Hol(M)$ acts transitively on the unit sphere in $T_xM$.} \newpage {\bf \blue Berger's theorem} \theorem (Berger's theorem, 1955) Let $G$ be an irreducible holonomy group of a Riemannian manifold which is not locally symmetric. {\bf \red Then $G$ belongs to the Berger's list:} { \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{\bf \color[rgb]{0,0,0.6}Berger's list}\\[1mm] \hline \it Holonomy & \it Geometry\\[1mm] \hline $SO(n)$ acting on $\R^n$ & Riemannian manifolds\\[1mm] \hline $U(n)$ acting on $\R^{2n}$ & K\"ahler manifolds\\[1mm] \hline $SU(n)$ acting on $\R^{2n}$, $n>2$ & Calabi-Yau manifolds\\[1mm] \hline $Sp(n)$ acting on $\R^{4n}$ & hyperk\"ahler manifolds\\[1mm] \hline $Sp(n)\times Sp(1)/\{\pm 1\}$ & quaternionic-K\"ahler\\[1mm] acting on $\R^{4n}$, $n>1$ & manifolds\\[1mm] \hline $G_2$ acting on $\R^7$ & $G_2$-manifolds \\[1mm] \hline $Spin(7)$ acting on $\R^8$ & $Spin(7)$-manifolds\\[1mm] \hline \end{tabular} } \remark There is one more group acting transitively on a sphere: $Spin(9)$ acting on $S^{15}\subset \R^{16}$. In 1968, D. Alekseevsky has shown that {\bf \purple a manifold with holonomy $Spin(9)$ is automatically locally symmetric.} {\small \remark A similar list exists for non-orthogonal irreducible holonomy without torsion (Merkulov, Schwachh\"ofer, 1999).} \end{document} \end{document}