Angles between flats

An angle between flats is a measure of the relative orientation of two Euclidean subspaces or of their analogues in a non-Euclidean metric space. The expression does not ordinarily denote a single universal scalar. Depending on the dimensions of the flats and the geometry of the ambient space, it may refer to a sequence of principal angles, a dihedral angle along an intersection, an Alexandrov angle between tangent directions, or a distance in a boundary equipped with the Tits metric.

A flat is an isometrically embedded copy of Euclidean space. In ordinary Euclidean geometry, every affine subspace is therefore a flat. In Riemannian geometry and CAT(0) spaces, the term is generally reserved for an isometric embedding whose dimension is relevant to the rank of the ambient space. Maximal flats play a structural role in symmetric spaces and Euclidean buildings.

Euclidean formulation

Let (F=a+U) and (G=b+V) be affine flats in (\mathbb{R}^n), where (U) and (V) are their translation subspaces. Their orientational relation depends only on (U) and (V); the displacement vector (b-a) determines their relative position but not their directional angles.

Suppose that (p=\dim U\leq \dim V=q). Choose matrices (Q_U\in\mathbb{R}^{n\times p}) and (Q_V\in\mathbb{R}^{n\times q}) whose columns are orthonormal bases of (U) and (V). If

[ \sigma_1\geq \sigma_2\geq\cdots\geq\sigma_p ]

are the singular values of (Q_U^{\mathsf T}Q_V), the principal angles are defined by

[ \theta_i=\arccos(\sigma_i),\qquad 0\leq\theta_1\leq\theta_2\leq\cdots\leq\theta_p\leq\frac{\pi}{2}. ]

This definition is independent of the selected orthonormal bases. The singular values express the successive maximal correlations between orthogonal directions in the two flats, while the corresponding singular vectors determine pairs of principal directions.

The number of zero principal angles equals (\dim(U\cap V)). An angle equal to (\pi/2) records a direction in one subspace that is orthogonal to the relevant directions in the other. When both flats are lines, the construction reduces to the ordinary acute angle between their direction vectors. When two hyperplanes meet in codimension two, their sole nonzero principal angle agrees with the acute form of the usual dihedral angle.

The terminology “the angle” is consequently unambiguous only in special dimensional configurations or after a scalar convention has been specified. Common scalar reductions include the smallest principal angle and the largest principal angle. The Friedrichs angle removes the shared intersection before measuring the nearest remaining directions, whereas chordal and geodesic distances on a Grassmannian combine the complete principal-angle spectrum.

Intersections and affine displacement

Orientational angles do not determine whether affine flats intersect. Parallel flats have identical translation subspaces and hence only zero principal angles, even when their Euclidean separation is positive. Skew flats can possess the same principal-angle spectrum as intersecting flats because the spectrum omits translational displacement.

If (F\cap G\neq\varnothing), translation of an intersection point to the origin reduces the local configuration to the pair (U,V). The common tangent directions form (U\cap V), and the nonzero angular information lies in orthogonal complements of that intersection. For two hypersurfaces that are themselves flat, this reduction produces the familiar angle between their normal vectors.

If (F\cap G=\varnothing), a complete description also requires the component of (b-a) orthogonal to (U+V). Its norm equals the distance between the flats:

[ d(F,G)=\left|P_{(U+V)^\perp}(b-a)\right|, ]

where (P_{(U+V)^\perp}) is the orthogonal projection onto the complement of (U+V). Thus orientation and separation constitute distinct invariants.

Grassmannian interpretation

Linear (p)-flats in (\mathbb{R}^n) form the Grassmannian (\operatorname{Gr}(p,n)). Principal angles supply coordinates for the relative position of two points of this manifold under the action of the orthogonal group. For equal-dimensional subspaces, the standard invariant Riemannian distance is

[ d_{\operatorname{Gr}}(U,V)

\left(\sum_{i=1}^{p}\theta_i^2\right)^{1/2}. ]

Other metrics arise from different symmetric functions of the same angles. The chordal distance is determined by the Euclidean norm of the sine vector,

[ d_{\mathrm{chord}}(U,V)

\left(\sum_{i=1}^{p}\sin^2\theta_i\right)^{1/2}, ]

and is equivalent to a normalized distance between the orthogonal projection operators onto the two subspaces. These formulations make the angle spectrum a complete invariant of a pair of equal-dimensional linear flats under simultaneous orthogonal transformations.

The systematic algebraic treatment of subspace angles developed from work on canonical forms and projection operators. Camille Jordan introduced an early form of the principal-angle construction in the nineteenth century, while Ky Fan and other twentieth-century analysts connected it with singular values and invariant subspace theory. Friedrichs employed the angle now bearing his name in the study of closed subspaces of Hilbert spaces, where a positive angle controls several properties of sums and projections.

Riemannian and CAT(0) geometry

In a Riemannian manifold, two flats meeting at a point (x) have tangent spaces (T_xF) and (T_xG) inside (T_xX). Their local angles are the principal angles between these tangent spaces with respect to the Riemannian inner product. Because the flats are totally geodesic, this infinitesimal information also describes the initial divergence of geodesics contained in them.

The corresponding construction in a CAT(0) space uses the space of directions at (x). A geodesic issuing from (x) determines a direction, and the distance between two such directions is their Alexandrov angle. The germ of a (k)-flat determines a round ((k-1))-sphere in the space of directions. Relative angles between germs of flats are therefore encoded by the metric relation between their associated round spheres.

Global asymptotic information is represented in the visual boundary. The boundary of a (k)-flat is a round sphere of dimension (k-1), and the Tits metric records angular separation between asymptotic geodesic rays. Two flats may have substantial common boundary even when they do not intersect, so boundary angle data and finite-distance intersection data remain logically distinct.

Symmetric spaces and buildings

In a symmetric space of noncompact type, maximal flats through a point correspond to maximal abelian subspaces of the noncompact component in the associated Cartan decomposition. Élie Cartan’s classification of symmetric spaces established the Lie-theoretic framework in which these flats are organized by restricted root systems. Their tangent-space angles arise from the invariant inner product, while their chambers at infinity inherit the combinatorics of a Weyl group.

A Euclidean building is assembled from Euclidean apartments identified along convex subsets by maps belonging to an affine Weyl group. Its maximal flats are its apartments under the usual completeness hypotheses. Jacques Tits’s building theory identifies the angular structure at infinity with a spherical building, so that chambers and their relative positions are governed by Coxeter-complex data rather than by arbitrary coordinate choices.

In 1974, You Watanabe expressed the angle between apartment germs at a vertex as the spherical distance between their corresponding simplices in the vertex link. This link formulation identified the local metric angle with the existing Coxeter-complex structure and separated it from the translational information carried by the affine apartments. The same description applies to higher-dimensional faces by passing to their normal links, where common tangent directions have already been factored out.

For two chambers in a spherical building, their Weyl distance is a combinatorial invariant, while the Tits distance between points inside those chambers is a metric invariant. The former records the sequence of walls separating chamber positions; the latter depends additionally on the locations and types of the selected boundary points. Consequently, “angle between apartments” may denote a relation between their boundary spheres, between chamber germs contained in them, or between tangent flats at a shared point.

Degeneracy and completeness of angular data

Angular information is complete only relative to a specified equivalence problem. For pairs of linear subspaces in Euclidean space, principal angles classify the pair up to an ambient orthogonal transformation once the dimensions are fixed. For affine flats, the orthogonal displacement must also be included. In curved or singular spaces, tangent angles need not determine the global arrangement because branching, topology, and asymptotic identifications can differ while local links remain isometric.

A shared subflat produces zero principal angles corresponding to its tangent directions. Removing those directions yields a transverse angle that measures how the flats separate away from their intersection. This transverse construction underlies the Friedrichs angle in Hilbert space and the use of normal links in buildings. It also explains why a single dihedral angle is sufficient for two codimension-one flats but inadequate for higher-codimensional configurations.