In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle.
Read more about Double Tangent Bundle: Secondary Vector Bundle Structure and Canonical Flip, Canonical Tensor Fields On The Tangent Bundle, (Semi)spray Structures, Nonlinear Covariant Derivatives On Smooth Manifolds, See Also
Famous quotes containing the words double and/or bundle:
“I met Jack Kennedy in November, 1946.... We went out on a double date and it turned out to be a fair evening for me. I seduced a girl who would have been bored by a diamond as big as the Ritz.”
—Norman Mailer (b. 1923)
““There is Lowell, who’s striving Parnassus to climb
With a whole bale of isms tied together with rhyme,
He might get on alone, spite of brambles and boulders,
But he can’t with that bundle he has on his shoulders,
The top of the hill he will ne’er come nigh reaching
Till he learns the distinction ‘twixt singing and preaching;”
—James Russell Lowell (1819–1891)