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Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.

Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a canonical fashion. An Ehresmann connection is a choice of horizontal subbundle $H \subseteq TTM$ which is complimentary to $V$, in that the double tangent bundle admits the horizontal decomposition $TTM = V \oplus H$.

One may define an Ehresmann connection by way of a connection form $v$.1 This is a bundle homomorphism $v : TTM \to TTM$ which satisfies $v^2 = v$ and $\operatorname{im}(v) = V$, and this generates the horizontal subbundle $H = \operatorname{ker}(v)$. One should think of $v$ as projecting onto the vertical subspace along $H$.

Suppose we are given an Ehresmann connection $H$ and connection form $v$. I would like to use these to generate a semispray. A semispray is a vector field on $TM$ (i.e., a section of $TTM$) which satisfies a certain compatibility condition with the tangent structure, and should somehow be compatible with the connection. I can see from Wikipedia how a semispray generates a torsion-free Ehresmann connection, but it is not clear to me how to use an Ehresmann connection (possibly with torsion) to generate a semispray.

1. The space $\mathcal C$ of connection forms is the subspace of $TTM$-valued $1$-forms $\Omega^1(TM, TTM)$ which satisfy $v^2 = v$ and $\operatorname{im}(v) = V$. Is there a concise, common name for the space $\mathcal C$? Does it have nice algebraic or topological structure?

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Isn't an Ehresmann connection on a tangent bundle just the same as an affine connection? – Deane Yang Apr 15 '12 at 21:36
@Deane: affine connections satisfy an additional linearity property that Ehresman connections do not need to satisfy. Holonomy for an Ehresmann connection can be a non-linear diffeomorphism of the tangent space. – Ryan Budney Apr 15 '12 at 21:41
In other words, an Ehresmann connection "forgets" the linear structure of the fiber of the tangent bundle. Right? – Deane Yang Apr 15 '12 at 22:13
@Tom, can you clarify whether you're talking about Ehresmann connections on the total space of the tangent bundle or on the total space of the frame bundle? In your first line you've called $T\mathcal{M}$ the frame bundle which I'm guessing is a typo, but in your footnote you use $F\mathcal{M}$... – Paul Reynolds Apr 16 '12 at 18:34
Tom, that was for affine connections. An Ehresmann connection is oblivious to the linear structure of the tangent bundle or the group structure of the frame bundle, so there is not necessarily any way to use an Ehresmann connection on one to induce an Ehresmann connection on the other. – Deane Yang Apr 17 '12 at 13:32
up vote 2 down vote accepted

Isn't the spray associated to the connection just the geodesic differential equation? Let $\pi_M : TM \to M$ be bundle projection, $\pi_{TM} : TTM \to TM$ bundle projection.

A double tangent vector $w \in TTM$ represents parallel transport of $\pi_{TM}(w)$ if $v(w) = 0$. i.e. if $w$ is horizontal. You can identify the horizontal spaces with the tangent spaces of $M$ by taking the derivative of $\pi_M : TM \to M$. ($V$ is the kernel of these derivatives).

So take as your vector field on $TM$ the function $f : TM \to TTM$ where $\pi_{TM}(f(x))=x$ for all $x \in TM$ and $f(x)$ is the unique horizontal vector in $TTM$ such that $D\pi_M(f(x)) = x$.

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Thanks for the answer, Ryan. I'm looking for a more direct approach using $H$ and/or $v$. My context is of random connections, which really means I'm looking at a probability measure $\mathbb P$ over the space $\mathcal C$ of all connection forms. I am hoping for a more structural answer like, "the semispray is the image of $v$ under the map \Phi : \mathcal C \to \Gamma(TTM) ..." or "is the unique $f$ which satisfies the equation $v = ...$". – Tom LaGatta Apr 15 '12 at 19:27
I don't really understand your request. My description of $f$ is as the unique map satisfying an equation. In which way is this not the kind of structural you're looking for? – Ryan Budney Apr 15 '12 at 20:10
In particular, the so-called space of connection forms is defined only if you first fix a choice of connection. Then the difference between any other connection and the fixed connection is a connection form. So if you know the spray for the fixed connection, it is straightforward to find a formula for the spray of any other connection relative to the fixed spray using the connection form. – Deane Yang Apr 15 '12 at 21:33
@Deane, in this more general situation isn't the space of connection forms $\mathcal{C}$ (as defined in the tiny footnote) a torsor for the vector space of $\mathcal{V}$-valued $1$-forms on the total space whose kernels contain $\mathcal{V}$? I've got a feeling you were talking about covariant derivative operators on a vector bundle... This says something about its topological structure but I doubt it has much nice geometrical structure with this level of generality. – Paul Reynolds Apr 16 '12 at 12:22
Paul, thanks for clarifying this. – Deane Yang Apr 17 '12 at 13:09

The paper where this is developed as thoroughly as one may desire is Grifone's Structure presque tangente et connexions I See Proposition I.38 in page 306.

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It seems that the "reconstruction" of the semispray from its induced non-linear connection in the Wikipedia-page is in fact a construction of a compatible semispray from any given non-linear connection. Of course the torsion of the connection is lost in the process v -> H -> v_H.

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