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Let $M$ be a differentiable manifold, let $TM$ be its frame 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 $TFM$-valued TTM$-valued$1$-forms$\Omega^1(FM, TFM)$\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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# How do we use an Ehresmann connection to define a semispray?

Let $M$ be a differentiable manifold, let $TM$ be its frame 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 $TFM$-valued $1$-forms $\Omega^1(FM, TFM)$ 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?