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This discussion of maps and vector bundles covers the pull-back construction; pull-backs and Grassmannians; pull-back of differential forms and push-forward of vector fields; invariant forms and vector fields on Lie groups; the exponential map on a matrix group; the exponential map and right/left invariance on Gl(n; ℂ) and its subgroups; and immersion, submersion, and transversality.

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.

This chapter examines principal bundles, which is defined as the Lie group analog of a vector bundle. It covers principal bundles constructed from vector bundles; examples of Lie group quotients; cocycle construction examples; pull-backs of principal bundles; reducible principal bundles; and associated vector bundles.

This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant derivative on a given vector bundle E → M with fiber 𝕍ⁿ = ℝⁿRn or ℂⁿ. It looks at principal bundles and connections; connections and covariant derivatives; and horizontal lifts. The chapter gives an application to the classification of principal G-bundles up to isomorphism and explains connections, covariant derivatives, and pull-back bundles.

“Restoring Abundant Earth” elaborates on pulling back, involving large-scale protection of land and seas—the best way to save biodiversity. Drawing on Earth system science and ecological writings, this chapter opens with a narrative of biological abundance as an intrinsic property of life on Earth. For humanity to inhabit a biosphere that is a plenum of biological wealth, we must pull back from large swathes of nature so that life may thrive free from exploitation and untimely extinctions. Three frameworks are offered. The first is a current initiative, gaining visibility and support, called “Nature Needs Half.” It argues for conservation of at least 50 percent of representative ecosystems to staunch biodiversity losses, preempt mass extinction, and mitigate climate change. The second borrows from Roderick Nash’s “Island Civilization”: instead of humanity being the sea within which relics of wild nature are islands, wild nature must be the expanse within which human habitats are nestled. The third framework builds on “bioregionalism,” as an ecologically and socially desirable form of human inhabitation inspired by indigenous ways of life. Pulling back is neither retreat nor diminishment. It will enable a high quality of life sourced from the material and spiritual gifts of a living planet.

Just as there are vector spaces over ℂ, there are vector bundles whose fibres can be consistently viewed as ℂⁿ for some n. This chapter first defines these objects and then provides number of examples. The discussions cover complex vector bundles; complex bundles over surfaces in ℝ³; the tangent bundle to a surface in ℝ³; bundles over 4-dimensional submanifolds in ℝ⁵; complex bundles over 4-dimensional manifolds; complex Grassmannians; the exterior product construction; algebraic operations; and pull-back.