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The subject of envelopes has been part of differential geometry from the beginning. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized and unified. A key step in this process is the use of “generalized immersions”. These have been described elsewhere but, briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions “with singularities.” Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion solving the family that has a universal mapping property relative to all other full rank solutions. The beauty of this approach becomes apparent in the “Envelope Theorem”. With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.

Partial differential equations and differential geometry come together in the idea of a generalized immersion. This concept, defined by means of Grassmann bundles and contact forms, allows for ?immersions? with ?singularities.? Sophus Lie?s generalized solutions to partial differential equations are an important special case. The classical second fundamental form has a natural generalization in the context of generalized immersions. The rank of the form is then meaningful. A constant rank assumption on the generalized second fundamental form leads to a natural foliation of the generalized immersion, at least when the ambient space is a space of constant curvature. Questions about the total geodesy and regularity of the foliation are also addressed.

The space of bi-invariant affine connections is determined for arbitrary compact Lie groups. In particular, there is a surprising new family of such connections on SU(n)

The space of left invariant affine connections is determined for the compact irreducible Riemannian symmetric spaces.

The inner and outer automorphism groups of a Lie group are generalized by considering automorphisms in the category of homogeneous principal bundles. These automorphisms are then used to produce certain invariant gauge transformations of such bundles. Some aspects of the resulting action on the space of invariant connections are also described.

The model quench front problem is the following nonlinear eigenvalue problem; Determine a number $p > 0$ and a function $u(x,y)$ such that $u_x + u_{xx} + pu_{yy} = 0$ in the domain $0 \leqq y \leqq 1, - \infty u_c ,u(0,0) = u_c $, where $0 0$. The parameters B and $u_c $, are supposed given. Upper and lower bounds are obtained for p using a variant of the method of separation of variables. The method of lines for this problem is also discussed.

The fact that Lie algebra homomorphisms can be extended to Lie group homomorphisms, provided the source is simply connected, is well-known. The situation in the non-simply connected case is less clear. In this paper, we show how ideas from differential geometry provide a unifying viewpoint about such extension problems. In particular, the holonomy bundle of a flat invariant connection plays a central role.