We initiate the study of average intersection theory in real Grassmannians. We define the expected degree of the real Grassmannian as the average number of real k-planes meeting nontrivially random subspaces of , all of dimension , where these subspaces are sampled uniformly and independently from . We express in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed and ,
where denotes the degree of the corresponding complex Grassmannian and is monotonically decreasing with . In the case of the Grassmannian of lines, we prove the finer asymptotic
The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set of dimension its Chow hypersurface , consisting of the k-planes A in whose projectivization intersects X. Denoting , we show that
where each is of dimension , the expectation is taken with respect to independent uniformly distributed and denotes the m-dimensional volume of .
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: BU 1371/2-2
Funding statement: The first author was partially supported by DFG grant BU 1371/2-2.
A.1 Proof of Theorem 3.2
Since the density of A is invariant under the orthogonal group , we can assume that B is spanned by the first l standard vectors, i.e.,
where 1 is the identity matrix, is an matrix and is an matrix. (Again, abusing notation, we denote by A and B also matrices whose columns span the corresponding spaces.) If we sample A with i.i.d. normal Gaussians, the corresponding probability distribution for the span of its columns is invariant, and consequently it coincides with the uniform distribution. In order to compute the principal angles between A and B using (3.1), we need to orthonormalize the columns of A. Defining
we see that the span of the columns of A and is the same, and the columns of are orthonormal. The cosines of the principal angles between A and B are the singular values of the matrix
The coincide with the square roots of the eigenvalues of the positive semidefinite matrix
which are the same as the eigenvalues of (eigenvalues are invariant under cyclic permutations). Consider the Cholesky decomposition
Then the eigenvalues of N equal the eigenvalues of
We use now some facts about the multivariate Beta distribution (see [37, Section 3.3]). By its definition, the matrix U has a distribution, and [37, Theorem 3.3.4] states that the joint density of the eigenvalues of U is given by
Recall now that for , which implies the change of variable
A.2 Proof of Lemma 4.3
We begin with a general reasoning. Assume that . The unit normal vectors of at A, up to a sign, are uniquely determined by A. Lemma 4.1 provides an explicit description for them as follows. Let and be the orthonormal bases given by Lemma 3.1 for A and B, respectively; note that since . In particular, for , where are the principal angles between A and B. Let with . According to Lemma 4.1, the unit vector spans the normal space of at A. (We use here the representation of elements of and its tangent spaces by vectors in ; cf. Section 2.1.)
Fix now and recall that the functions give the smallest and second smallest principal angle, respectively, between and B. Let and put
We first prove that .
For we consider the curve for . We note that arises from A by a rotation with the angle t in the (oriented) plane spanned by and f, fixing the vectors in the orthogonal complement spanned by . We have and . It is well known (see, e.g., ) that is the geodesic through A with speed vector ν, that is, and . From the definition of the ε-tube, we therefore have
Since both and f are orthogonal to , the principal angles between and B are , where . By our assumption , we have . Hence we see that is the smallest principle angle between and B if . We have thus verified that .
For the other inclusion, let ; assume that , otherwise we clearly have . Let and be orthonormal bases of C and B, respectively, as provided by Lemma 3.1. So we have for all j, where and by the assumption . In particular, is orthogonal to and are linearly independent. We define as the space spanned by . By construction, and for . Hence, . Let be defined as above as the space resulting from A by a rotation with the angle t in the oriented plane spanned by and . By construction, we have , where . Therefore, we indeed have by (A.2).
A.3 Proof of Lemma 4.5
More generally, we consider a semialgebraic set of dimension . Consider the semialgebraic set
together with the projections on the two factors: and . As , the set is semialgebraic. Note that is compact if Y compact, and is connected if Y is connected (since is continuous). In order to determine the dimension of , we note that the fiber over is isomorphic to . As a consequence, we get
The fibers over consist of exactly one point, except for the A lying in the exceptional set
Note that consists of one point only, for all .
In order to show that , we consider the semialgebraic set
with the corresponding projections and . Note that . For such that , we have
since the fibers of are isomorphic to . Therefore,
and we see that . For the projection defined by , we have , hence
Moreover, the projection , is bijective, hence
Using , we see that
In the special case where , we conclude that is a hypersurface.
We consider now the following set of “bad” :
We claim that this semialgebraic set has dimension strictly less than . This follows for from (A.6), for from (A.5) applied to , for from Proposition 2.2 and (A.5), and finally for by Proposition 2.2.
Thus generic points are not in and hence satisfy the following:
The intersection consists of one point only (let us denote this point by p),
The point p is a smooth point of X,
The point A is a regular point of ,
The point is a regular point of .
It remains to prove that for every we have (4.3). To this end, let us take with . We work in local coordinates on a neighborhood U of in , where . For simplicity we center the coordinates on the origin, so that are the coordinates of . In this coordinates, the set can be described as
where represents the reduced local equations describing the condition and the reduced local equations giving the condition Since is a regular point of , the tangent space of at is described by
Note that p is a smooth point of X, hence is the equation for the tangent space to X at p. On the other hand, since , we have
Let now be the linear space corresponding to and let us write the equations for in the same coordinates as above (by construction we have ):
Note that the same equation as in (A.7) appears here (recall that this is the equation describing ), but now is replaced with its linearization at zero. In particular,
which coincides with (A.8). Since , this finally implies
which finishes the proof.
A.4 Proof of Proposition 5.12
We extend the function f to by setting
denoting it by the same symbol. Similarly, we extend g by setting
We assume now that has i.i.d. standard Gaussian entries. Then we can write
where denotes the joint density of the ordered singular values of X. This density can be derived as follows. The joint density of the ordered eigenvalues of the Wishart distributed matrix is known to be [37, Corollary 3.2.19]
From this, using and the change of variable , we obtain
As a consequence, we obtain
where in the second line we have switched to polar coordinates with and . Note that the power of the r-variable arises as
It is immediate to verify that
which completes the proof.
A.5 Generalized Poincaré’s formula in homogeneous spaces
The purpose of this subsection is to prove the kinematic formula in homogeneous spaces for multiple intersections and to derive Theorem 3.19. The proofs are similar to , to which we refer the reader for more details.
A.5.1 Definitions and statement of the theorem
In the following, G denotes a compact Lie group with a left and right invariant Riemannian metric. See  for background on Lie groups. We denote by the identity element and by the left translation by . The derivatives of will be denoted by . By assumption, this map is isometric.
In (3.5) we defined a quantity for capturing the relative position of linear subspaces of a Euclidean vector space. We can extend this notion to linear subspaces in tangent spaces of G at any points , assuming . This is done by left-translating the to the identity e: so we define
Let now be a closed Lie subgroup and denote by the quotient map. We endow K with the Riemannian structure induced by its inclusion in G, and with the Riemannian structure defined by declaring p to be a Riemannian submersion. For example, when with the invariant metric defined in Section 2.8 and , then with the quotient metric is isometric to the Grassmannian with the metric defined in Section 2.1.
Note that G acts naturally by isometries on ; if and , we denote by gy the result of the action. Further, we denote by the projection of the identity element. The multiplication with an element fixes the point ; as a consequence, the differential of k induces a map denoted , so that we have an induced action of K on .
Given a submanifold X of a Riemannian manifold M, we denote by NX its normal bundle in M (i.e., for all the vector space is the orthogonal complement to in ). Also, the restriction of the Riemannian metric of M to X allows to define a volume density on X; if is an integrable function, we denote its integral with respect to this density by .
For given submanifolds , we define the function
as follows. For let be such that for all i. We define
where the expectation is taken over a uniform .
The reader should compare this definition with [26, Definition 3.3], which is just a special case. The main result of this section is the following generalization of Poincaré’s kinematic formula for homogeneous spaces, as stated in [26, Theorem 3.8] for the intersection of two manifolds. We provide a proof, since the more general result is crucial for our work and we were unable to find it in the literature.
Let be submanifolds of such that
Then, for almost all , the manifolds intersect transversely, and
where the expectation is taken over a uniform .
Under the assumptions of Theorem A.2, if moreover G acts transitively on the tangent spaces to for , then we have
where is any point of
Let us look now at the special case and . If are coisotropic hypersurfaces of , then G acts transitively on their tangent spaces by Proposition 4.6. Moreover, when , it is easy to check that the constant value of equals the real average scaling factor defined in Definition 3.18. Hence in this case, the statement of Corollary A.3 coincides with the statement of Theorem 3.19.
A.5.2 The kinematic formula in G
As before, G denotes a compact Lie group. We derive first Theorem A.2 in the special case , which is the following result (we can without loss of generality assume ).
Let be submanifolds of G such that
In the special case of intersecting two submanifolds, this is an immediate consequence of the following “basic integral formula” from [26, Section 2.7] (take ).
Let be submanifolds of G such that
For almost all , the manifolds and intersect transversely, and if h is an integrable function on , then
where is the function given by .
In order to reduce the general case to that of intersecting two submanifolds, we first establish a linear algebra identity.
For subspaces of a Euclidean vector space, we have
We may assume that , since otherwise both sides of the identity are zero. Let us denote by , and orthonormal bases of , W and Z, respectively. Moreover, we denote by an orthonormal basis for obtained by completing . By definition we have
On the other hand, since , we have
Let be submanifolds of G and let be linear subspaces of tangent spaces of G (possibly at different points) such that
Then we have
We apply Proposition A.5 with the function defined by
When and intersect transversely, we have , which implies for . Hence we obtain with Proposition A.5,
where the last equality is due to Lemma A.6. ∎
Proof of Lemma A.4.
which completes the proof. ∎
Proof of Theorem A.2.
We consider , which is a submanifold, since the projection is a submersion, cf. [9, Theorem A.15]. Moreover, intersect transversally if do so. We can rewrite the integral in the statement as
For justifying the last equality, note that the coarea formula [9, Theorem 17.8] yields
for any submanifold Y of , since p is a Riemannian submersion. Applying Lemma A.4 to the integral in the last line, we obtain
The projection defined by
is a Riemannian submersion with fibers isometric to . Using the coarea formula
where the last equality is due to Definition A.1. (Note that here is where we use the right invariance of the metric under the action of the elements in K, as one can verify by a careful inspection of the last steps. The reader can see the proof of [26, Theorem 3.8], which is almost identical and where all the details of the calculation are shown). This completes the proof. ∎
We are very grateful to Frank Sottile who originally suggested this line of research. We thank Paul Breiding, Kathlén Kohn, Chris Peterson, and Bernd Sturmfels for discussions. We also thank the anonymous referees for their comments.
 D. Amelunxen, Geometric analysis of the condition of the convex feasibility problem, PhD thesis, University of Paderborn, Paderborn 2011. Search in Google Scholar
 S. Basu, A. Lerario, E. Lundberg and C. Peterson, Random fields and the enumerative geometry of lines on real and complex hypersurfaces, preprint (2016), https://arxiv.org/abs/1610.01205. 10.1007/s00208-019-01837-0Search in Google Scholar
 T. Bröcker and T. Tom Dieck, Representations of compact Lie groups, Grad. Texts in Math. 98, Springer, New York 1995. Search in Google Scholar
 J. Draisma and E. Horobeţ, The average number of critical rank-one approximations to a tensor, Linear Multilinear Algebra 64 (2016), no. 12, 2498–2518. 10.1080/03081087.2016.1164660Search in Google Scholar
 J. Draisma, E. Horobeţ, G. Ottaviani, B. Sturmfels and R. R. Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math. 16 (2016), no. 1, 99–149. 10.1007/s10208-014-9240-xSearch in Google Scholar
 A. Edelman, T. A. Arias and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20 (1999), no. 2, 303–353. 10.1137/S0895479895290954Search in Google Scholar
 Y. V. Fyodorov, A. Lerario and E. Lundberg, On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys. 95 (2015), 1–20. 10.1016/j.geomphys.2015.04.006Search in Google Scholar
 D. Gayet and J.-Y. Welschinger, Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 105–120. 10.1112/jlms/jdu018Search in Google Scholar
 D. Gayet and J.-Y. Welschinger, Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 4, 733–772. 10.4171/JEMS/601Search in Google Scholar
 I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Mod. Birkhäuser Class., Birkhäuser, Boston 2008. Search in Google Scholar
 A. Gray and L. A. Vanhecke, The volumes of tubes in a Riemannian manifold, Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), no. 3, 1–50. Search in Google Scholar
 J. Harris, Algebraic geometry, Grad. Texts in Math. 133, Springer, New York 1995. Search in Google Scholar
 S. L. Kleiman, The transversality of a general translate, Compos. Math. 28 (1974), 287–297. Search in Google Scholar
 E. Kostlan, On the distribution of roots of random polynomials, From topology to computation: Proceedings of the Smalefest (Berkeley 1990), Springer, New York (1993), 419–431. 10.1007/978-1-4612-2740-3_38Search in Google Scholar
 S. E. Kozlov, Geometry of real Grassmannian manifolds. I, II, III, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 246 (1997), 84–107, 108–129, 197–198. Search in Google Scholar
 A. Lerario and E. Lundberg, Gap probabilities and Betti numbers of a random intersection of quadrics, Discrete Comput. Geom. 55 (2016), no. 2, 462–496. 10.1007/s00454-015-9741-7Search in Google Scholar
 S. Lojasiewicz, Ensembles semi-analytiques, Institut des Hautes Études Scientifiques, Bures-sur-Yvette 1965. Search in Google Scholar
 L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts Monogr. 6, American Mathematical Society, Providence 2001. Search in Google Scholar
 D. Mumford, Algebraic geometry. I. Complex Projective Varieties, Grundlehren Math. Wiss. 221, Springer, Berlin 1976. Search in Google Scholar
 L. A. Santaló, Integral geometry and geometric probability, Encyclopedia Math. Appl., Addison–Wesley, Reading 1976. Search in Google Scholar
 R. Schneider, Convex bodies: The Brunn–Minkowski theory, expanded ed., Encyclopedia Math. Appl. 151, Cambridge University Press, Cambridge 2014. Search in Google Scholar
 M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice 1992), Progr. Math. 109, Birkhäuser, Boston (1993), 267–285. 10.1007/978-1-4612-2752-6_19Search in Google Scholar
 F. Sottile, Enumerative real algebraic geometry, Algorithmic and quantitative real algebraic geometry (Piscataway 2001), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 60, American Mathematical Society, Providence (2003), 139–179. 10.1090/dimacs/060/11Search in Google Scholar
 M. Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Wilmington 1979. Search in Google Scholar
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