How to learn intrinsic torsion

Question

I want to learn about G-structure and intrinsic torsion. But I can find no textbook that details it. If you can give me a reference about it, it would be much appreciated.

Answer 1

General $G$-structures are not usually treated in textbooks, other than the basic definitions, so it's not surprising that there would be few if any textbooks that treat the intrinsic torsion of $G$-structures. However, it's a straightforward concept based on a simple question: For a given $G$-structure, what is the obstruction to $G$ being flat to first order at every point?

For example, Riemannian metrics in dimension $n$ are $\mathrm{O}(n)$-structures on $n$-manifolds, so $G=\mathrm{O}(n)$. It's a famous result that every Riemannian metric is flat to first order at every point (this is essentially the Fundamental Lemma of Riemannian Geometry), so the intrinsic torsion of every $G$-structure vanishes in this case.

For $G=\mathrm{U}(n)\subset\mathrm{SO}(n)$, a $G$-structure on $M^{2n}$ is what is known as an almost Hermitian structure on $M^{2n}$, i.e., a choice of an almost complex structure $J$ and a compatible metric $g$, which defines a canonical $2$-form $\omega$ such that $\omega(v,w) = g(Jv,w)$. In this case, the intrinsic torsion of the $G$-structure turns out to be composed of two tensors: the $3$-form $\mathrm{d}\omega$ and the Nijnhuis tensor $N_J$ of the almost complex structure $J$. If they both vanish, the $G$-structure is said to be Kähler.

Of course, there is a systematic approach that defines the intrinsic torsion of $G$-structures on $n$-manifolds for any Lie subgroup $G\subset\mathrm{GL}(n,\mathbb{R})$. It is implicit in the works of Élie Cartan (who introduced the notion of what we now call $G$-structures). It is made explicit in the later works of Chern, Ehresmann, Spencer and many others. The basic point is that, given a $G$-structure $B$ on $M^n$, there is a naturally defined subbundle ${\frak{g}}_B\subset TM\otimes T^*M$, which induces a natural skew-symmetrizing mapping $$ \delta: {\frak{g}}_B\otimes T^*M\to TM\otimes\Lambda^2(T^*M). $$ The cokernel of $\delta$, denoted $H^{0,1}_B(M)$, is a vector bundle over $M$ that is associated to the principal $G$-bundle $B$. The intrinsic torsion $\tau_B$ of $B$ is a section of this bundle, and $\tau_B$ vanishes at a point $x\in M$ if and only if $B$ is flat to first order at $x$.

One can find explicit formulae in the works cited above, including computations of the cokernel of $\delta$ for various $G$ of interest. The recipe for $\tau_B$ is the following: Given a $G$-structure $B$, let $\nabla_0$ be any connection on $M$ that is compatible with $B$. Then $T(\nabla_0)$, the torsion of $\nabla_0$, is a section of $TM\otimes \Lambda^2(T^*M)$. If $\nabla_1$ is any other connection on $M$ that is compatible with $B$, the difference $\nabla_1-\nabla_0$ is a section of ${\frak{g}}_B\otimes T^*M$. So $T(\nabla_1)-T(\nabla_0)$ is a section of $\delta\bigl({\frak{g}}_B\otimes T^*M\bigr)$. Hence $\tau_B = \bigl[T(\nabla_0)\bigr]$, as a section of $H^{0,1}_B(M)$, does not depend on the choice of $\nabla_0$. This is the intrinsic torsion of $B$. Almost by definition, it's the part of the torsion of a $B$-compatible connection that is independent of the choice of such a connection.

It's important to note that, because $H^{0,1}_B(M)$ is defined as a quotient of a tensor bundle, when $G$ is not reductive it may not be possible to interpret $H^{0,1}_B(M)$ as a subbundle of some tensor bundle of $M$ (i.e., some natural subbundle of the tensor product of some number of copies of $TM$ with some other number of copies of $T^*M$). Thus, one cannot always interpret the intrinsic torsion as some collection of tensors on $M$ in the classical sense. This probably contributes to some confusion in the literature when people call the intrinsic torsion a 'tensor' even though it's not always interpretable as a tensor in the classical sense.

Answer 2

A standard published reference (so you can cite it) which contains the definitions that Robert wrote out in his answer:

Joyce, Dominic D.(4-OX) Riemannian holonomy groups and calibrated geometry. Oxf. Grad. Texts Math., 12 Oxford University Press, Oxford, 2007. x+303 pp. ISBN:978-0-19-921559-1, pp. 36-39, section 2.6.