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May be $\pi:Y \mapsto X$ a general fibred manifold. Is it true that in fibred manifold a connection always exists? This wiki article states this: https://en.wikipedia.org/wiki/Connection_(fibred_manifold)

Connection is an additional structure on a manifold and the manifold has to be smooth as a requirement that a connection exists. Or can connections be defined also in non-smooth manifolds? Which manifolds have neither a metric nor a connection (any examples?)?

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You can just take a differentiable metric on the total space of the fibred bundle and define the connection to be the distribution orthogonal to the fibres.

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