Scalar Curvature Estimates by Parallel Alternating Torsion

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500Comment: 17 page

The Command and Control of Canadian and American Maritime Air Power in the Northwest Atlantic, 1941-1943

Operational, organizational, doctrinal, and cultural differences hampered effective command and control of Canadian and American maritime air power defending shipping against U-boats off the east coast during the Second World War. The American desire to implement US unity of command over both nations’ maritime air forces clashed with the Canadian preference for simple cooperation. Canadian airmen resisted several American attempts to impose unity of command until the operational situation in the Battle of the Atlantic revealed inefficiencies in coordination which necessitated all maritime air power in the Northwest Atlantic be centralized under Canadian operational control in the spring of 1943

Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.Comment: LaTeX, 40 pages; v2: references added, BL = IK announced; v3: reference to DWW = IK adde

Scalar curvature estimates for compact symmetric spaces

We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.Comment: 13 pages, LaTeX, uses amsar

Vafa-Witten Estimates for Compact Symmetric Spaces

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.Comment: LaTeX, 11 pages. V2: Rigidity statement added, minor changes. To appea