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A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank. (English) Zbl 1271.53039

Let \(S_+(p,n)\) denote the set of real square matrices of order \(n\) that are symmetric, positive semidefinite and of fixed rank \(p\). This set can be endowed with the Euclidean metric by considering it as a subset of \(\mathbb R^{n\times n}\). Even though this metric seems to be natural for \(S_+(p,n)\), it is known that geodesics are not always complete.
In this paper the authors endow \(S_+(p,n)\) with a Riemannian metric \(\tilde{g}\) which turns it into a complete homogeneous space. Indeed, it is isometric to the quotient manifold \(\text{GL}^+(n,\mathbb R)/\text{Stab}_{E}\). Here, \(\text{GL}^+(n,\mathbb R)\) represents the group of matrices having positive determinant which is equipped with a left-invariant metric and \(\pi:\text{GL}^+(n,\mathbb R)\longrightarrow \text{GL}^+(n,\mathbb R)/\text{Stab}_{E}\) is a Riemannian submersion. Several explicit formulas concerning the geometry of \((S_+(p,n),\tilde{g})\) are given and it is proved that this manifold is complete as a consequence of the study of horizontal geodesics on \(\text{GL}^+(n,\mathbb R)\).
To conclude, the authors compare advantages and disadvantages of this homogeneous Riemannian metric with those previously considered on \(S_+(p,n)\).

MSC:

53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
53C22 Geodesics in global differential geometry
15B48 Positive matrices and their generalizations; cones of matrices
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