User meneldur - MathOverflow

Do you set a one or two commas when using \mapsto?

2013-05-06T09:34:58Z

I am currently revising a paper and I am completely confused about the commas. Is it correct English to write

1) "The canonical map $X \to Y$, $x \mapsto f(x)$, is injective."

or is it

2) "The canonical map $X \to Y$, $x \mapsto f(x)$ is injective." ,

i.e. is the second comma mandatory, optional or wrong?

Topology of the Universal Spinor Field Bundle

2012-11-25T17:33:24Z

While reading article [1] below I came across the notion of a universal spinor bundle. This is defined at the beginning of section 6 (p.14) in [1] as follows: Let $M$ be a spin manifold and $\mathcal{G}$ be a set of smooth-semi Riemannian metrics that is open in the $\mathcal{C}^\infty$-topology (for simplicity let $\mathcal{G} = \mathcal{R}(M)$, the set of all Riemannian metrics). Define $E_{(g,x)} :=\Sigma^g_x M$ to be the spinor space at $x \in M$ with respect to $g \in \mathcal{R}(M)$. The author claims that

i) The map $\pi:E \to \mathcal{R}(M) \times M$, $\psi \in \Sigma^g_xM \mapsto (g,x)$ is a fibre bundle.

ii) The space $\mathcal{S}_g$ of sections (probably smooth sections?) of $\pi^{-1}({g} \times M)$ is a Frechet manifold.

iii) These spaces assemble to a Frechet fibre bundle $\mathcal{S}:=\bigcup_{g \in \mathcal{R}(M)} \to \mathcal{R}(M)$.

Prior to these claims the author refers the reader to [2,p.153ff] for more details. But unfortunately, I can't find a proof of these claims in there. Probably [2] is supposed to be a general introduction to Lagrangian field theory, which is an important subject in the rest of the section. I am however interested in the Universal Spinor Bundle in its own right. Therefore this raises the following

Question: How are (i)-(iii) proven? More explicitely I am asking

1) How exactly are the spaces $E$, $\mathcal{S}_g$ and $\mathcal{S}$ topologized? Equivalently, how do local trivializations look like and why are their transition functions continuous resp. smooth?

2) Are $\mathcal{S}_g$ really the smooth sections and can this construction be generalized to $L^2$-sections?

3) Can someone give a reference for more details on the Universal Spinor bundle?

4) Why does one not consider $E$ as a bundle over $\mathcal{R}(M)$? Here the later I would give the $\mathcal{C}^1$-topology (this is commmon in spin geometry).

Possible Solution: I thought about it for a while and came across the idea that the identification of the spinor bundles with different metrics as discussed in [1, Section 5] or [2, Section 2] could be helpful to construct local trivializations. But I am not sure what formal argument to use in order to show that they depend continuously on the metrics or in what sense one should define continuity here. I am also unsure, if this way of thinking is not way too complicated.

[1] Bär, Gauduchon, Moroianu - Generlized Cylinders in Semi-Riemannian and Spin Geometry, http://arxiv.org/abs/math/0303095

[2] Deligne. Quantum fields and strings

[3] Maier: Generic Metrics and Connection on Spin- and Spin-c-manifolds

Integration of differential forms using measure theory?

2011-01-02T13:36:38Z

Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{A}:=\sigma(\tau_M)$ is the $\sigma$-Algebra generated by the topology $\tau_M$ of $M$ and for any $A \in \mathcal{A}$, $\mu(A):=\int_M{\chi_A d_gV}$, where $\chi_A:M \to [0,1]$ is the characteristic function of $A$. We obtain $\int_{M}{f d\mu} = \int_M{f d_gV}$, where the left hand side is understood to be an integral in the measure theoretic sense and the right hand side is an integration of a density. This enables us to define the space $L_p(\mu)$ with norm $\|f\|_{L_p(M)}^p = \int_{M}{|f|d_gV}$ on a manifold and apply all the results from integration theory to it, e.g. that it is a Banach space and so on.

My question is: Does this work in the following more general setup: Extend the Riemannian metric on $M$ to a fibre metric in $\bigwedge^k T^{\;*}M$, $0 \leq k \leq m$, (as described in the paragraph below). Then one may define $L_p$-spaces of differential forms by setting $\|\omega\|_{L_p(M)}^p := \int_{M}{|\omega|^pd_gV}$ and setting $L_p^k(M)$ to be the space of all measurable $k$-forms on $M$ (i.e. with Lebesgue measurable coefficient functions in any chart) such that $\|\omega\|_{L_p(M)}<\infty$. Is it possible to construct a measure space $(M,\mathcal{A},\mu)$ such that $L_p^k(M)$ may be thought of as an $L_p(\mu)$ as well?. The problem obviously is the range of a differential form. Formally it is a map $\omega\colon M \to \bigwedge^k T^*M$, i.e. it takes values in the vector bundle $\bigwedge^kT^*M$. Even if integration theory is available for functions on measure spaces with values in Banach spaces, this does not help since the bundle itself is not a vector space. I am interested in this question, because otherwise I see no alternative but to establish all the results about integration theory for $L_p^k(M)$ again, i.e. that it is a Banach space, Lebesgue Dominated Convergence Theorem, Fubini/Tonelli etc. That seems a bit exaggerated since intuitively this space is not so fundamentally different.

Construction of the fibre metric: For any $0 \leq k \leq m$ the Riemannian metric may be extended canonically to differential forms in $\Omega^k(M)$ in the following way: For one forms $\omega,\eta \in \Omega^1(M)$ define $g(\omega,\eta):=g(\omega^\sharp, \eta^\sharp)$, where $\sharp:T^*M \to TM$ is the sharp operator with respect to $g$. Then define $g$ on decomposable forms by $g(\omega^1 \wedge \ldots \wedge \omega^k, \eta^1 \wedge \ldots \wedge \eta^k):= \det(g(\omega^i, \eta^j))$.

Answer by Meneldur for Integration of differential forms using measure theory?

2011-01-03T13:30:18Z

Thanks for your posts. I am summing up what we have got so far.

@Dmitri Pavlov: You explained why the answer to my question is negative and give an alternative approach to $L_p$-spaces on hermitian vector bundles in your answer. You claim that all the usual theorems of measure theory hold in this more general setting. Can you give a reference for that?

@Deane Yang: You claim that this can be done locally. That idea seems natural to me, but I'm afraid I cant make this rigorous: Let $E \to M$ be a real vector bundle of rank $k$ with fibre metric $h$ and let $(M,g)$ be a Riemannian $m$-manifold. Assume $U \subset M$ is open and sufficiently small such that there exists a chart $\varphi:U \to R^m$ and a local trivialization $\Psi=(\Psi_1,\Psi_2):\pi^{-1}(U) \to U \times R^k$. Let $\mu_g$ be the Riemannian volume density on $M$, $\tau_g:=\sqrt{\det(g_{ij})}$ and $\left\| \cdot \right\|_h$ be the norm induced by $h$. Then for any section $s \in \Gamma(E)$, we obtain

$$\int_{U}{\left\| s \right\|_h^p \mu_g} = \int_{V}{(\left\| s \right\|_h^p \tau_g \circ \varphi^{-1} )(x) dx}=\int_{V}{\left\| s(\varphi^{-1}(x)) \right\|^p_{h(\varphi^{-1}(x))} \tau_g(\varphi^{-1}(x)) dx }$$

Now of course $\tilde s:= \Psi_2 \circ s\circ \varphi^{-1}:V \to R^k$ is a vector valued function, but you can't choose a norm $\left| \cdot \right|$ on $R^k$ such that for any $x \in V$, we obtain $\left| \tilde s(x) \right| = \left\| s(\varphi^{-1}(x) \right\|_{h(\varphi^{-1}(x))} $, because the fibre metric may change in every point.

Comment by Meneldur

2012-11-29T09:47:47Z

Also an interesting idea. Maybe someone knows a suitable reference to Fréchet manifold theory?

Comment by Meneldur

2012-11-26T08:39:38Z

Thanks for your answer. I knew this paper before and had a look at it again now. Granted, Bourguignon-Gauduchon as well as Maier provide a map $\beta_{g,h}:\Sigma^g M \to \Sigma^h M$ between the two spinor bundles with respect to two fixed Riemannian metrics $g,h$. Since $\beta_{g,h}$ is an isomorphism, it is of course continuous in its argument. They also construct a Hilbert space isomorphism $\bar \beta_{g,h}:L^2(\Sigma^gM) \to L^2(\Sigma^hM)$ between the associated spaces of $L^2$-sections. But I am asking why and in what sense these constructions are continuous in the metrics $g,h$?

Comment by Meneldur

2011-01-02T17:19:23Z

I know it is not rigorous. I indeed mean that $L_p^k(M)$ is isometrically isomorphic to $L_p(\mu)$, but if $L_p(\mu)$ is only defined for real-valued functions this cannot be true. So this implicitely contains the question if $L_p(\mu)$ makes sense for functions $f:\Omega \to E$, were $E$ is a vector bundle of finite rank rather than a vector space.