 By Hitchin N.

Best differential geometry books

The topology of fibre bundles

Fibre bundles, now an essential component of differential geometry, also are of serious value in sleek physics--such as in gauge thought. This booklet, a succinct creation to the topic via renown mathematician Norman Steenrod, was once the 1st to provide the topic systematically. It starts with a normal creation to bundles, together with such issues as differentiable manifolds and overlaying areas.

Surgery on simply-connected manifolds

Brower W. surgical procedure on simply-connected manifolds (Springer, 1972)(ISBN 0387056297)

The Geometry of Total Curvature on Complete Open Surfaces

This autonomous account of recent rules in differential geometry indicates how they are often used to appreciate and expand classical ends up in imperative geometry. The authors discover the impression of overall curvature at the metric constitution of entire, non-compact Riemannian 2-manifolds, even supposing their paintings will be prolonged to extra common areas.

Differential Geometry, Lie Groups, and Symmetric Spaces, Volume 80

The current booklet is meant as a textbook and reference paintings on 3 subject matters within the name. including a quantity in growth on "Groups and Geometric research" it supersedes my "Differential Geometry and Symmetric Spaces," released in 1962. for the reason that that point a number of branches of the topic, rather the functionality thought on symmetric areas, have built considerably.

Extra info for Differentiable manifolds

Example text

3 the cohomology class [α0 ] is non-zero. e. that α = cα0 + dγ. 64 Given α use the partition of unity to write α= ϕi α then by linearity it is sufficient to prove the result for each ϕi α, so we may assume that the support of α lies in a coordinate neighbourhood Um . e. renumbering, we have p ∈ U1 , Ui ∩ Ui+1 = ∅, q ∈ Um . Now for 1 ≤ i ≤ m − 1 choose an n-form αi with support in Ui ∩ Ui+1 and integral 1. 1, there is a form β0 with support in U1 such that α0 − α1 = dβ1 . Continuing, we get α0 − α1 α1 − α2 ...

The unit ball {x ∈ Rn : x ≤ 1} is a manifold with boundary S n−1 . 3. The M¨obius band is a 2-dimensional manifold with boundary the circle: 4. The cylinder I × S 1 is a 2-dimensional manifold with boundary the union of two circles – a manifold with two components. 59 We can define differential forms etc. on manifolds with boundary in a straightforward way. Locally, they are just the restrictions of smooth forms on some open set in Rn to (Rn )+ . A form on M restricts to a form on its boundary.

46 Proof: Consider the right hand side RX (α) = d(i(X)α) + i(X)dα. Now i(X) reduces the degree p by 1 but d increases it by 1, so RX maps p-forms to p-forms. Also, d(d(i(X)α) + i(X)dα) = di(X)dα = (di(X) + i(X)d)dα because d2 = 0, so RX commutes with d. Finally, because i(X)(α ∧ β) = i(X)α ∧ β + (−1)p α ∧ i(X)β d(α ∧ β) = dα ∧ β + (−1)p α ∧ dβ we have RX (α ∧ β) = (RX α) ∧ β + α ∧ RX (β). On the other hand ϕ∗t (dα) = d(ϕ∗t α) so differentiating at t = 0, we get LX dα = d(LX α) and ϕ∗t (α ∧ β) = ϕ∗t α ∧ ϕ∗t β and differentiating this, we have LX (α ∧ β) = LX α ∧ β + α ∧ LX β.