Genericity of nondegenerate geodesics with general boundary conditions. (English) Zbl 1213.58012

Bettiol, Renato G.; Giambò, Roberto

Topol. Methods Nonlinear Anal. 35, No. 2, 339-365 (2010).

Let \(M\) be a possibly noncompact manifold of dimension \(n\). For each \(\nu\in\{0,\dots,n\}\), consider the set \(\text{Met}_{\nu}^{k}(M)\) of all semi-Riemannian \(C^{k}\) metrics of index \(\nu\), which is a subset of \(\Gamma_{\text{sym}}^{k}(T^{*}M\otimes T^{*}M)\), the set of \(C^{k}\) sections \(s\) such that \(s_{x}:T_{x}M\times T_{x}M\to\mathbb R\) is symmetric for all \(x\in M\).
If \(M\) is compact, \(\Gamma_{\text{sym}}^{k}(T^{*}M\otimes T^{*}M)\) has a natural Banach space structure, and \(\text{Met}_{\nu}^{k}(M)\) is an open subset. In order to endow the space of tensors over a noncompact manifold \(M\) with a Banach space structure the authors, consider a vector subspace \(\mathcal E\) of \(\Gamma_{\text{sym}}^{k}(T^{*}M\otimes T^{*}M)\), called a \(C^{k}\)-Whitney type Banach space of tensor fields over \(M\), satisfying the conditions:

(a)

\(\mathcal E\) contains all tensor fields having compact support;

(b)

\(\mathcal E\) is endowed with a Banach space norm \(\|\cdot\|_{\mathcal E}\) with the property that \(\|\cdot\|_{\mathcal E}\)-convergence of a sequence implies convergence in the weak Whitney \(C^{k}\)-topology.

Using an auxiliary Riemannian metric \(g_{\text{R}}\) on \(M\), it is possible to construct a \(C^{k}\)-Whitney type Banach space of tensors on \(M\). Such a space \(\mathcal E\) will be fixed in the sequel. Let \(\mathcal A_{\nu}\subset\mathcal E\cap\text{Met}_{\nu}^{k}(M)\) be an open subset of the intersection.
Let \(\mathcal M\) be the product \(M\times M\). For all semi-Riemannian metrics \(g\) on \(M\) define \(\bar g=g\oplus(-g)\) on \(T\mathcal M=TM\oplus TM\). For all submanifolds \(\mathcal P\subset \mathcal M\), define \(\mathcal A_{\nu,\mathcal P}=\{g\in\mathcal A_{\nu}\,:\, \mathcal P \text{ is nondegenerate with respect to }\bar g=g\oplus(-g)\}\). If \(\mathcal P\) is compact, then \(\mathcal A_{\nu,\mathcal P}\) is open in \(\mathcal A_{\nu}\). A \(\nu\)-general boundary condition on \(M\) is a compact submanifold \(\mathcal P\subset M\times M\) such that \(\mathcal A_{\nu,\mathcal P}\) is nonempty.
It is well-known that the set \(H^{1}([0,1],M)\) of all curves of Sobolev class \(H^{1}\) in \(M\) has a canonical Hilbert manifold structure. The subset \[ \Omega_{\mathcal P}(M)=\{\gamma\in H^{1}([0,1],M)\,:\, (\gamma(0),\gamma(1))\in\mathcal P\} \] is a submanifold of \(H^{1}([0,1],M)\). Fix \(g\in\mathcal A_{\nu,\mathcal P}\). A \(g\)-geodesic \(\gamma\in\Omega_{\mathcal P}(M)\) is a \((g,\mathcal P)\)-geodesic if it satisfies \((\dot\gamma(0),\dot\gamma(1))\in (T_{(\gamma(0),\gamma(1))}\mathcal P)^{\perp}\), where \(\perp\) denotes orthogonality relative to \(\bar g\).
A \((g,\mathcal P)\)-geodesic \(\gamma\) is a critical point of the generalized energy \(C^{k}\)-functional \(f:\mathcal A_{\nu,\mathcal P}\times\Omega_{\mathcal P}(M)\to\mathbb R\) defined by \(f(g,\gamma)=\frac{1}{2}\int_{0}^{1}g(\dot\gamma,\dot\gamma)\,dt\). A \((g,\mathcal P)\)-geodesic is nondegenerate if it is a nondegenerate critical point of \(f\).
A \(\nu\)-general boundary condition \(\mathcal P\) is admissible if, for every \(g_{0}\in\mathcal A_{\nu,\mathcal P}\), there exists an open neighbourhood \(\mathcal V\) of \(g_{0}\) in \(\mathcal A_{\nu,\mathcal P} \) and \(a>0\), such that for all \(g\in\mathcal V\), an every \((g,\mathcal P)\)-geodesic \(\gamma\), the \(g_{\text{R}}\)-length \(L_{\text{R}}(\gamma)\geq a\).
With this definitions the main result of the paper is the following:
Theorem 5.10: Let \(M\) be a smooth \(n\)-dimensional manifold and \(\nu\in\{0,\dots,n\}\) an index. Fix \(\mathcal E\subset\Gamma_{\text{sym}}^{k}(T^{*}M\otimes T^{* }M)\) a \(C^{k}\)-Whitney type Banach space of tensor fields over \(M\) and \(\mathcal A_{\nu}\subset\mathcal E\cap\text{Met}_{\nu}^{k}(M)\) an open subset. Consider \(\mathcal P\) an admissible \(\nu\)-general boundary condition. Then the following is a generic subset in \(\mathcal A_{\nu,\mathcal P}\)
\[ \mathcal G_{\mathcal P}(M)=\{g\in \mathcal A_{\nu,\mathcal P}\,:\, \text{ all } (g,\mathcal P)\text{-geodesics }\gamma\in\Omega_{\mathcal P}(M)\text{ are nondegenerate}\}. \]
The paper ends with an extension of the above genericity in the \(C^{\infty}\)-context. Renaming \(\mathcal A_{\nu,\mathcal P}\) and \(\mathcal G_{\mathcal P}(M)\) as \(\mathcal A_{\nu,\mathcal P}^{k}\) and \(\mathcal G_{\mathcal P}^{k}(M)\) to stress the dependence on \(C^{k}\)-regularity of tensor fields, the authors show that \(\mathcal G_{\mathcal P}^{\infty}(M):=\bigcap_{k\in\mathbb N}\mathcal G_{ \mathcal P}^{k}(M)\) is generic in \(\mathcal A_{\nu,\mathcal P}^{\infty}=\bigcap_{k\in\mathbb N}\mathcal A_{\nu,\mathcal P}^{k}\) in the \(C^{\infty}\)-topology.

Reviewer: David Marin Perez (Bellaterra)

MSC:

58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57R70	Critical points and critical submanifolds in differential topology
58D17	Manifolds of metrics (especially Riemannian)

Keywords:

generic properties; semi-Riemannian geodesic flow; nondegenerate geodesics; general endpoints conditions