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:
“In a symbol there is concealment and yet revelation: here therefore, by silence and by speech acting together, comes a double significance.... In the symbol proper, what we can call a symbol, there is ever, more or less distinctly and directly, some embodiment and revelation of the Infinite; the Infinite is made to blend itself with the Finite, to stand visible, and as it were, attainable there. By symbols, accordingly, is man guided and commanded, made happy, made wretched.”
—Thomas Carlyle (1795–1881)
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)