Comparison theory for Riccati equations
Let E be a finite dimensional vector space with inner product \(<, >\), and let S(E) be the vector space of self-adjoint linear endomorphisms of E. For a given smooth curve R: \({\mathbb{R}}\to S(E)\) the authors consider solutions B: (0,t\({}_ 0)\to S(E)\) of the matrix Riccati differential equation \((R)\quad B'(t)+B^ 2(t)+R(t)=0.\) This is closely related to the matrix Jacobi differential equation \((J)\quad X''(t)+R(t)X(t)=0.\) Indeed, whereever it is defined \(B(t)=X'(t)X^{-1}(t)\) is a solution of (R) of X(t) is a solution to (J). Both equations are important in Riemannian geometry. For example, equation (R) is the evolution equation for the shape operators of a family \(\{M_ t\}\) of parallel hypersurfaces if R(t) denotes the curvature transformation determined by the normal vector to \(M_ t.\) The main result of this paper is an elementary proof of the following theorem, which improves an earlier result by the first author: Theorem. Let \(R_ 1,R_ 2: {\mathbb{R}}\to S(E)\) be smooth with \(R_ 1\geq R_ 2\). For \(j=1,2\) let \(B_ j: (0,t_ j)\to S(E)\) be a solution of \((R_ j)\) with maximal \(t_ j\in (0,\infty]\), such that \(U:=B_ 2-B_ 1\) has a continuous extension to 0 with U(0)\(\geq 0\). Then \(t_ 1\leq t_ 2\) and \(B_ 1\leq B_ 2\) on \((0,t_ 1)\). Moreover, \(d(t):=\dim \ker U(t)\) is monotonically decreasing on \((0,t_ 1)\). In particular, if \(B_ 1(s)=B_ 2(s)\) for some \(s\in (0,t_ 1)\), then \(B_ 1=B_ 2\) and \(R_ 1=R_ 2\) on [0,s]. In this result the continuity of U at 0 is convenient but not necessary, and one may replace the condition U(0)\(\geq 0\) by the condition \(\lim_{t\to 0} \inf u(t)\geq 0,\) where u(t) is the smallest eigenvalue of U(t). The authors also describe the behavior at \(t=0\) of a solution of (R): Proposition. Let B: (0,t\({}_ 0)\to S(E)\) be a solution of (R). Then there is a projection \(P\in S(E)\), \(P^ 2=P\), such that \(C(t):=B(t)-P/t\) has a continuous extension to \(t=0\). Moreover, \(im P\subseteq C(0)\).
scientific article; zbMATH DE number 4175916