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This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary, cornered asymptotically hyperbolic manifolds, and proves a theorem of Cartan–Hadamard-type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary.

In this paper, we derive a local Hamilton type gradient estimate for a nonlinear heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem.

In this note we show that the moduli space M(Sg,n) of surface Sg,n of genus g with n punctures, satisfying 3g + n ≥ 5, admits no complete Riemannian metric of nonpositive sectional curvature such that the Teichmuler space T(Sg,n) is a mapping class group Mod(Sg,n)-invariant visibility manifold.

Let $(M,g)$ be an $n$-dimensional compact boudaryless Riemannian manifold with nonpositive sectional curvature, then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions to smooth submanifolds of dimension $k$, for $p>\dfrac{2n}{n-1}$ when $k=n-1$ and $p>2$ when $k\leq n-2$, compared to the general results of Burq, G��rard and Tzvetkov \cite{burq}. Earlier, B��rard \cite{Berard} gave the same improvement for the case when $p=\infty$, for compact Riemannian manifolds without conjugate points for $n=2$, or with nonpositive sectional curvature for $n\geq3$ and $k=n-1$. In this paper, we give the improved estimates for $n=2$, the $L^p$ norms of the restrictions of eigenfunctions to geodesics. Our proof uses the fact that, the exponential map from any point in $x\in M$ is a universal covering map from $\mathbb{R}^2\backsimeq T_{x}M$ to $M$, which allows us to lift the calculations up to the universal cover $(\mathbb{R}^2,\tilde{g})$, where $\tilde{g}$ is the pullback of $g$ via the exponential map. Then we prove the main estimates by using the Hadamard parametrix for the wave equation on $(\mathbb{R}^2,\tilde{g})$, the stationary phase estimates, and the fact that the principal coefficient of the Hadamard parametrix is bounded, by observations of Sogge and Zelditch in \cite{SZ}. The improved estimates also work for $n\geq 3$, with $p>\frac{4k}{n-1}$. We can then get the full result by interpolation., 17 pages. arXiv admin note: text overlap with arXiv:1108.2726

A singular foliation on a complete Riemannian manifold M M is said to be Riemannian if each geodesic that is perpendicular to a leaf at one point remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previous results of the authors on singular Riemannian foliations with sections.

We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio 1 must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio 1 must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.

This is a continuation of the Cambridge Tract “Harmonic maps between Riemannian polyhedra”, by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

A class of diffusion processes on the path space over a compact Riemannian manifold is constructed. The diffusion of such a process is governed by an unbounded operator. A representation of the associated generator is derived and the existence of a certain local second moment is shown.

We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold ( M , g ) (M,g) with boundary ∂ M \partial M whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics g ′ g’ on M M there is a C 3 , α C^{3,\alpha } -neighborhood of g g such that g g is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance — as measured in M M ). More precisely, given any metric g ′ g’ in this neighborhood with the same boundary distance function there is diffeomorphism φ \varphi which is the identity on ∂ M \partial M such that g ′ = φ ∗ g g’=\varphi ^{*}g . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.

In this paper, we study noncompact complete Riemannian nmanifolds with n > 3 which are not pointwise conformal to subdomains of any compact Riemannian n-manifold. For this, we compare the Sobolev Quotient at infinity of a noncompact complete Riemannian manifold with that of the singular set in a compact Riemannian manifold using the method for the Yamabe problem.

We find conditions for the exponential map on a weak riemannian Hilbert manifold to be a nonlinear Fredholm map of index zero and apply the result to left-invariant Sobolev metrics on loop groups.

We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion.

The soul theorem states that any open Riemannian manifold (Mg) with nonnegative sectional curvature contains a totally geodesic compact submanifold S such that M is diffeomorphic to the normal bundle of S. In this paper we show how to modify g into a new metric g' so that: 1. g' has nonnegative sectional curvature and soul S. 2. The normal exponential map of S is a diffeomorphism. 3. (M, g') splits as a product outside of a compact set. As a corollary we obtain that any such M is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.

In this paper we show that on a complete Riemannian manifold of negative curvature and dimension $n>1$ every two points which realize a local maximum for the distance function are connected by at least $2n+1$ geometrically distinct geodesic segments (i.e. length minimizing). Using a similar method, we obtain that in the case of non-positive curvature, for every two points with the same property as above the number of connecting distinct geodesic segments is at least $n+1$., 14 pages, AMS-LaTex

We relate the growth rate of volume in the universal cover of a compact Riemannian manifold to the growth in the fundamental group in terms of word length in a given set of generators and the length of geodesics representing these generators.

Our main goal is to determine the general solution of the functional equation \[ f 1 ( x y ) + f 2 ( x y − 1 ) = f 3 ( x ) + f 4 ( y ) + f 5 ( x ) f 6 ( y ) , f i ( t x y ) = f i ( t y x ) ( i = 1 , 2 ) \begin {array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_3}(x) + {f_4}(y) + {f_5}(x){f_6}(y),} \\ {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \\ \end {array} \] where f i {f_i} are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao’s quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d’Alembert and Wilson equations.

We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if f : M → X f:M \to X is a continuous surjection from a smooth manifold to a compact metric space with f − 1 ( x ) {f^{ - 1}}(x) connected for every x ∈ X x \in X , then there is a metric d on X and a sequence of Riemannian metrics { ψ i } \{ {\psi _i}\} on M so that ( M , ψ i ) (M,{\psi _i}) converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function ρ \rho and a sequence of Riemannian manifolds with ρ \rho as contractibility function so that lim ( M , ψ i ) \lim (M,{\psi _i}) is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function ρ \rho so that for every ε > 0 \varepsilon > 0 there are Riemannian metrics ϕ ε {\phi _\varepsilon } and ψ ε {\psi _\varepsilon } on M and N so that ( M , ϕ ε ) (M,{\phi _\varepsilon }) and ( N , ψ ε ) (N,{\psi _\varepsilon }) have contractibility function ρ \rho and d G H ( ( M , ϕ ε ) , ( N , ψ ε ) ) > ε {d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) > \varepsilon .

The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let h : [ 0 , 1 ] → T 0 M h:[0,1] \to {T_0}M be a C 1 {C^1} function where M is a compact Riemannian manifold, o ∈ M o \in M is a base point, and T o M {T_o}M is the tangent space to M at o ∈ M o \in M . Let W ( M ) W(M) be the space of continuous paths from [0,1] into M, ν \nu be Wiener measure on W ( M ) W(M) concentrated on paths starting at o ∈ M o \in M , and H s ( ω ) {H_s}(\omega ) denote the stochastic-parallel translation operator along a path ω ∈ W ( M ) \omega \in W(M) up to "time" s. (Note: H s ( ω ) {H_s}(\omega ) is only well defined up to ν \nu -equivalence.) For ω ∈ W ( M ) \omega \in W(M) let X h ( ω ) {X^h}(\omega ) denote the vector field along ω \omega given by X s h ( ω ) ≡ H s ( ω ) h ( s ) X_s^h(\omega ) \equiv {H_s}(\omega )h(s) for each s ∈ [ 0 , 1 ] s \in [0,1] . One should interpret X h {X^h} as a vector field on W ( M ) W(M) . The vector field X h {X^h} induces a flow S h ( t , ∙ ) : W ( M ) → W ( M ) {S^h}(t, \bullet ):W(M) \to W(M) which leaves Wiener measure ( ν ) (\nu ) quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if h ( 1 ) = 0 h(1) = 0 and the Wiener measure ( ν ) (\nu ) is replaced by a pinned Wiener measure ( ν e ) ({\nu _e}) . (The measure ν e {\nu _e} is proportional to the measure ν \nu conditioned on the set of paths which start at o ∈ M o \in M and end at a fixed end point e ∈ M e \in M .) Also as in [13], one gets an integration by parts formula for the vector-fields X h {X^h} defined above.

Quantitative versions of unique continuation are proved for finite sums of eigenfunctions of the Laplacian on compact Riemannian manifolds. The results include a lower bound for the order of vanishing, a growth estimate for the supremum on compact balls, and a gradient bound. For real analytic metrics, an upper bound for the Hausdorff measure of the zero set is derived.

We study the behavior of geodesics passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point.

In their article, P. Li and S. T. Yau give a lower bound of the first Neumann eigenvalue in terms of geometrical invariants for a compact Riemannian manifold with convex boundary. The purpose of this paper is to generalize their result to a compact Riemannian manifold with possibly nonconvex boundary.

We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold. Our object is to give a quick proof of a result usually obtained only for analytic manifolds using combinatorial manipulation with power series. Let Xf be a C()-manifold. Put VEC(,#) for the space of C*)-vector fields on X#. Given X E VEC(#) and p E X we write exp(tX)p for the point on the manifold corresponding to the flow of X at time t that passes through p at time t = O. Thus one has (d/dt)f o exp(tX)It= = f' o X everywhere on X, where f' is the derivative of the mapping f: # -:+ R. If exp(tX) is globally defined then (d/dt)f o exp(tX) = P o X o exp(tX) = f o T(exp(tX)) o X. Here we use the notation T(S): T(Xf) -T(df) for the functorial map of the tangent bundle corresponding to a C(9)-map S: X# X. Let S be an automorphism of Xf. Given V E VEC(4t) we get a new vector field (AdS)Vd T(S)oVoS1. (1) Proposition. Suppose X, V E VEc(A) and that exp X is globally defined. Then, for any test function f e C(?) () we have (d/dt)f o (exp(X + tV)) o exp(-X)J I = f' of Ad(expsX) Vds. Remark. X and V are fixed in the statement. Thus, for each p E X, s F-4 Ad(expsX) V(p) gives a continuous map [0, 1] -+ Tp(A#). f' is a linear functional on the finite-dimensional vector space Tp(.). Thus one has an Received by the editors June 6, 1990 and, in revised form, November 21, 1990. 1991 Mathematics Subject Classification. Primary 58A30, 22E30.

A point p in a Riemannian manifold M is weakly spherical if for each point q ≠ p there is either exactly one or at least three minimizing geodesic segments joining p to q. In this note, it is shown that round 2-dimensional spheres are the only Riemannian surfaces with a weakly spherical point realizing the injectivity radius.

We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian. Let (Mn, g) be a compact, connected Riemannian manifold of n dimensions. The Laplacian Ag acting on functions on M has discrete spectrum. Let A1 (g) denote the smallest positive eigenvalue of Ag . Hersch [5] proved that AIl(g)vol(S2, g) 3, the sphere Sn admits metrics of volume one with A1 arbitrarily large [3, 6]. Bleecker conjectured in [3] that such metrics exist on every manifold Mn if n > 3. In this note we give a very simple proof of Bleecker's conjecture using known examples and quite general principles. The same result has been proved independently by Xu [7] by a construction similar to ours. His argument, however, is much harder than our proof. Theorem 1. Every compact manifold Mn with n > 3 admits metrics g of volume one with arbitrarily large AI (g) . Proof. The idea of the proof is very simple. We take a metric go on Sn with vol(Sn, go) = 1 and AL (go) > k + 1, where k is a large constant. We excise from Sn a very small ball B(p, t) = B. and form the connected sum of Sn with M. The resulting manifold is diffeomorphic to M and has a submanifold Q, with smooth boundary, naturally identified with Sn \ Bi . Let g1 be an Received by the editors February 10, 1993. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 53C21. This work was done while the second author enjoyed the hospitality of Forschungsinstitut fur Mathematik at ETH Zurich. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

Let (M, g) be a compact, connected, CO Riemannian manifold of n dimensions. Denote by RI ,p(M, g) the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree p. We prove that for n > 4 and 2 < p < n 2, there exists a family of metrics gt of volume one, suchthat AI ,p(M,gt) -+oo as t -+oo.

The main part of this note is to show a general covering lemma in Rn, n 2 2 , with the aim to obtain the estimate for BMO norm and the volume of a nodal set of eigenfunctions on Riemannian manifolds. This article is a continuation of our previous work [L]. In [L] we proved a covering lemma in R2 and applied it to the BMO norm estimates for eigenfunctions on Riemannian surfaces. The principal part of this article is to prove a general covering lemma in Rn for n 2 2 . As applications, we can obtain the BMO estimate for eigenfunctions and the volume estimate for the nodal set. Let M n be a smooth, compact, and connected Riemannian manifold with no boundary. Let A denote the Laplacian on M n . Let -Au = ilu, u an eigenfunction with eigenvalue il , il > 1 . Our main results can be stated as follows Theorem A (BMO estimate for log lul) . For u , il as above and n 2 3 , where C is independent of il and u and is only dependent on n and M n Theorem B (geometry of nodal domains). Let n 2 3 and u , il as above. Let B c M n be any ball, and let R c B be any of the connected components of {x E B :u(x) # 0) . If R intersects the middle halfof B , then where C is independent of il and u Donnelly and Fefferman [DFl, DF2] and Chanillo and Muckenhoupt [CM] proved Theorem A with (10gil)~ replaced by iln(n+2)/4 and iln log2 , respectively, and Theorem B with A-2n2-n14 replaced by il-(n+n2(n+2))/2 and il-2n2-n/2 (log il)-2n , respectively. Received by the editors June 28, 1991. 1991 Mathematics Subject Class$cation. Primary 35B05, 58G03. @ 1993 American Mathematical Society 0002-9939193 $1.00 + $.25 per page

Let M M be a compact Riemannian manifold with H 1 ( M , Z ) = 0 {H_1}(M,Z) = 0 . We show that, for a point p ∈ M p \in M , the cut locus and conjugate locus of p p must intersect if M M admits a group of isometries which fixes p p and has principal orbits of codimension at most 2. This is a classical theorem of Myers [5] in the case when M M has dimension 2.

In this paper we develop an existence theory for minimal 2-spheres in compact Riemannian manifolds. The spheres we obtain are conformally immersed minimal surfaces except at a finite number of isolated points, where the structure is that of a branch point. We obtain an existence theory for harmonic maps of orientable surfaces into Riemannian manifolds via a complete existence theory for a perturbed variational problem. Convergence of the critical maps of the perturbed problem is sufficient to produce at least one harmonic map of the sphere into the Riemannian manifold. A harmonic map from a sphere is in fact a conformal branched minimal immersion. We prove the existence of minimizing harmonic maps in two cases. If N is a compact Riemannian manifold with j2(N) = 0, then every homotopy class of maps from a closed orientable surface M to N contains a minimiz

Let M M be a pseudo-Riemannian manifold. Locally a distance function may be defined. The bisector of two points is the set of points equidistant from these two points. Our main result is that the bisector of two points which are not zero distance apart is a totally geodesic submanifold of M M if and only if M M has constant curvature.

The nontangential maximal function associated to the Poisson semigroup for a compact Riemannian manifold is shown to be weak type (1,1) and LP bounded (p > 1). Let M be a compact Riemannian manifold with Riemannian measure dV and Laplace-Beltrami operator A. Let pt(x,y) be the fundamental solution to L = d 2/dt2 + A = 0; thus, for f continuous on M, Ptf(y) = fM pt(x, y)f(x) dV(x) satisfies LPtf = 0 and limt_0 Ptf(y) = f(y) for each y in M. It is a standard fact that when the class of boundary functions is suitably enlarged, almost everywhere convergence still holds. Indeed, a stronger theorem, proved by C. S. Herz in a more general context [3], is that the radial maximal function P. f = supt>o PtI ff is bounded on LP(M) for p > 1 and is of weak type (1,1) (cf. [4, p. 48] for the latter case). Let d(x, y) be the distance function on M associated to the Riemannian metric, let L.,(y) = {(y', t) E M x (0, oo) I d(y', y) 1) and of weak type (1,1). Standard technique then allows the conclusion of nontangential boundary convergence lim Ptf (y) = f (yo) dV-a.e. for f in LP(M) (p > 1), t-4+o (Y, t) Er (yo) although this result is otherwise easily obtainable by P.D.E. methods. Our method is to prove a two-sided inequality for the Poisson kernel pt (x, y) which immediately reduces our theorem to Herz's. This inequality is obtained by utilizing known behavior of the heat kernel in conjunction with the subordination of the Poisson semigroup to the heat diffusion semigroup, a special case of Bochner's principle of subordination. The author used these ideas in an earlier paper where M was either a compact semisimple Lie group (superseded by the current paper) or a symmetric space of noncompact type [1]; in both cases exact formulae for the heat kernel were used whereas in the present paper we use the Minakshisundaram-Pleijel asymptotic expansion. Received by the editors July 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 58G20, 43A32; Secondary 42B25, 31C12. Research supported in part by NSF grant DMS 8402202. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

We derive the Gauss curvature equation and the Codazzi-Mainardi equation with respect to a semisymmetric metric connection on a Riemannian manifold and the induced one on a submanifold. We then generalize the theorema egregium of Gauss.

We consider maps, from one Riemannian manifold to another, which are critical for all invariantly defined functionals on the space of maps. There are many such critical mappings, perhaps too numerous to suitably classify, although a characterization of sorts is provided. They are proven to have constant rank, with the image being a homogeneous minimal submanifold of the target manifold. Critical maps need not be Riemannian submersions onto their images. Also, there are homogeneous spaces for which the identity map is not critical. Many open problems remain.

We make use of the variational properties of the geodesic distance function of a Riemannian manifold and the technique of the "blowing-up" (a-process) on a complex manifold to derive the nonexistence of compact complex analytic subvarieties in a simply connected, complete Hermitian manifold with nonpositive sectional curvature. 1. Notation and preliminaries. In this section we introduce the notation and definitions that will be used in the following sections. (i) Let Mn be a complete connected Riemannian manifold of dimension n and class C '. Throughout this note we will indicate by r = the geodesic distance function of M. MA = the diagonal of the product manifold M X M. Mcut = the set of all cut pairs of M. M V M = M x M \ MA U mcut Wc= {(p, q) E M X M/r(p, q) = c), for every positive real number c. IIW = the second fundamental form of Wc (we observe that Wc n M V M is a submanifold of M V M of codimension 1) with grad r taken as the normal vector to Wc. For every (p, q) in M V M we introduce a natural orthonormal frame of M V M at (p, q) as follows: Let y be the unique geodesic (parametrized by arc-length) in M between p and q and let {ei'(p), 1 < i < n) be an orthonormal frame of M at p, with ej(p) = the tangent vector to y at p; by parallel translation along y from p to q we construct an orthonormal frame {ei"(q), 1 < i < n) of M at q and then we set e?1(p, q) = \k [(e'(p) 0) ? (0, ei"(q))], 1 iSn. We introduce the subspaces n n V(p,q)= Rei(p, q) and V(,q)= 2 Re_(p q) i=l i=l Received by the editors May 23, 1978. AMS (MOS) subject classifications (1970). Primary 53C55; Secondary 32C25.

For F = R , C F = {\mathbf {R}},{\mathbf {C}} or H {\mathbf {H}} an F F -axial map is defined to be an axial map R P m × R P m → R P m + k {\mathbf {R}}{P^m} \times {\mathbf {R}}{P^m} \to {\mathbf {R}}{P^{m + k}} equivariant with respect to diagonal and trivial F ∗ {F^{\ast }} -actions. Analogously to the real case, it is shown that C {\mathbf {C}} -axial maps correspond to immersions of C P n {\mathbf {C}}{P^n} in R 2 n + k {{\mathbf {R}}^{2n + k}} while (for F = R F = {\mathbf {R}} and for F = C F = {\mathbf {C}} , k k odd) embeddings induce F F -symmaxial maps. Examples are thereby given of symmaxial maps not induced by embeddings of R P n {\mathbf {R}}{P^n} , and of R {\mathbf {R}} -axial maps which are not C {\mathbf {C}} -axial. Furthermore, the relationships which hold when F = R , C F = {\mathbf {R}},{\mathbf {C}} are no longer valid for F = H F = {\mathbf {H}} .

If G G is a compact Lie group and M M a Riemannian G G manifold, then the orbit map ∏ : M → M / G \prod :M \to M/G is a stratified Riemannian submersion and the well-known "cohomogeneity method" pioneered by Hsiang and Lawson [HL] reduces the problem of finding codimension k k minimal submanifolds of M M to a related problem in M / G M/G . We show that this reduction of variables technique depends only on a certain natural Riemannian geometric property of the map ∏ \prod which we call h h -projectability and which is shared by certain other naturally occurring and important classes of Riemannian submersions.

For a general submanifold of a pseudo Riemannian manifold, the exponential mapping of the orthogonal bundle into the ambient manifold may fail to be a diffeomorphism on the zero section. Here we show that differential geometric information intrinsic to the submanifold determines when this map has a fold singularity. Throughout this paper /': N -* (M, ( )) will denote a C00 immersion of smooth manifolds of dimension « and m, respectively. M will come equipped with a smooth nondegenerate geodesically complete pseudo Riemannian metric. If N± denotes the orthogonal bundle over N, A/1- = {vq& TqM\i(p) = q, vq ± i*(TpN)}, then the geodesic tube map /: N1, -» M is defined by v -* e\pHp)v. If (M, ( >) is Riemannian (i.e. ( ) is positive definite), then / is a local diffeomorphism on a neighborhood of any point on the zero section of A7± . If ( M, ( )) is not definite this is not always the case. In particular, if Npx n TpN is a non trivial vector space then it is clear that (/')" cannot have full rank ato^eJV1. The purpose of this paper is to show that in the case where NJ- n T N is one-dimensional the intrinsic geometry of N (i.e. invariants associated with (N, i*( ))) determine when the geodesic tube map has a simple fold singularity at op (i.e. aS10 singularity; see Golubitsky and Guillemin, p. 87). The author encountered this phenomena in a geometric study of a special class of first order partial differential equations. There it is used to determine certain qualitative features of solutions. One may pose the problem of describing how the geometry of /: N -> (M, ( )) determines the structure of more complicated singularities in the geodesic tube map. Aside from shedding light upon some of the finer features of pseudo Riemannian geometry, the resolution of this matter would be of use in the study of the aforementioned first order partial differential equations. The author wishes to express thanks to R. Bryant, J. Damon, P. Eberlein, R. B. Gardner, M. Schlessinger and G. Thompson for inspiration. Our program is as follows. Because Npx n TpN # 0 implies that i*( )p is a degenerate bilinear form we begin by adapting the classical connection construction to such a setting. This is the dual connection. At a point p where ;*( > is degenerate the dual connection is used to define a conformai structure on T N. We show that this conformai structure determines when the degeneracy of i*( ) satisfies two

THEOREM 2. Assume J * V 0 . Writing UtQ=(je a o\0 )U&9 J(\ is the disjoint union of closed submanifolds. REMARK. If d i m M = 2 , e^J=^" 8 , and if d i m M = 3 , the hypothesis that 1F*J£0 can be dropped. The proof of Theorem 1 also allows us to conclude that a solution h of the linearized equations DR(g0) • h=0 is tangent to a curve of exact solutions of R(g)=p through a given solution g0, provided p is not a constant ^ 0 . In the terminology of [4] we say the equation R(g)=p is linearization-stable at g0. From Theorem 3 below the equation R(g)=0 is still linearization-stable about a solution g0 provided Ric(g0) is not identically zero. For the singular case p = 0 , Theorem 2 incorporates an isolation theorem inspired by the work of Brill and Deser [2], namely, that the flat metrics are isolated solutions of R(g)=0. As a corollary one has: If g(t) is a

Let ( P , g ) (P,g) be a compact, connected, C ∞ {C^\infty } Riemannian ( n + 1 ) (n + 1) -manifold ( n ⩾ 1 ) (n \geqslant 1) with a unit Killing vector field with dual 1 1 -form η \eta . For t > 0 t > 0 , let g t = t − 1 g + ( t n − t − 1 ) η ⊗ η {g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta , a family of metrics of fixed volume element on P P . Let λ 1 ( t ) {\lambda _1}(t) be the first nonzero eigenvalue of the Laplace operator on C ∞ ( P ) {C^\infty }(P) of the metric g t {g_t} . We prove that if d η d\eta is nowhere zero, then λ 1 ( t ) → ∞ {\lambda _1}(t) \to \infty as t → ∞ t \to \infty . Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which λ 1 {\lambda _1} is unbounded on the space of fixed-volume metrics.

Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers' condition) which ensure the compactness of a complete Riemamnan manifold. Using standard index form techniques we relate the problem of finding such Ricci curvature criteria to that of establishing the conjugacy of the scalar Jacobi equation. Using this relationship we obtain a Ricci curvature condition for compactness which is weaker than that of Ambrose and, in fact, which is best among a certain class of conditions. One of the most well-known results relating the curvature and topology of a complete Riemannian manifold M is the classical theorem of Myers [8] which states that if the Ricci curvature with respect to unit vectors on M has a positive lower bound then M is compact. (Myers also gives a diameter estimate in terms of this bound.) In 1957 Ambrose [1] published an interesting generalization of Myers' theorem. He proved that if there is a point q M in such that along each geodesic y: [0, oo) -M emanating from q (and parameterized by arc length t) the Ricci curvature satisfies f Ric(dY dtY ) dt = +oo then M is compact. One of the important features of this result is that the Ricci curvature is not required to be everywhere nonnegative. The author, together with T. Frankel, has applied Ambrose's theorem to certain problems in general relativity (see [4]). In this paper we present a general technique for establishing compactness criteria for complete Riemannian manifolds. As an application of this technique we obtain a generalization of Ambrose's theorem, which, with respect to a certain class of curvature conditions, is best. This generalization can be used to improve some of the results in [4]. (See, especially, Theorem 5 and Corollary 6 of that paper.) We take a moment to introduce some notation and terminology. Throughout, let M denote a smooth complete Riemannian manifold of dimension n > 2. Let be the Riemannian metric on M and let V be the associated Levi-Civita connection. If t -y(t) is a curve in M, let D/dt be the covariant derivative operator on vector fields along y induced by the connection V. For vector fields X and Y let R(X, Y) be the Riemann curvature transformation, i.e. R(X, Y)Z = VxV yZ-V yVxZ -V[XyZ Received by the editors October 3, 1980. 1980 Mathematics Subject Classificatio. Primary 53C20. o 1982 American Mathematical Society 0002-9939/82/0000-0025/$02.25

Lower and upper bounds for the smallest positive eigenvalue of the Laplacean on a parallelizable Riemannian manifold are combined to obtain an explicit lower bound for the torsion of global orthonormal frame fields in terms of the diameter of the metric. INTRODUCTION AND RESULTS Consider a compact, connected n-dimensional Riemannian manifold M with metric tensor g. We assume that the tangent bundle TM is trivial. Then there exist parallelizations ow: TM -* Rn mapping g into the euclidean metric on R . Such co will be called orthonormal. Equivalently, the vector fields El defined by w(E,) = e, = ith standard basis vector of R0 form an orthonormal frame field on M. The torsion T of X is defined by T(X, Y) = co (dwo(X, Y)) for vector fields X and Y on M. T is the torsion tensor of the flat connection D on TM defined by DEI = 0. It is well known that a compact Riemannian manifold admits a torsion-free orthonormal parallelization if and only if it is isometric to a flat torus. In this note we study the relation between isometric invariants of a Riemannian manifold and the torsion of its orthonormal frame fields. Specifically, let K denote the maximum norm of T, i.e., K = max{flT(X, Y)fl: X, Y E TyM, IfXWI = IYl= 1, x E M}. The smallest positive eigenvalue of the Laplace operator of (M, g) on functions is denoted by A1 . Theorem 1. Let (M, g) be a compact connected Riemannian manifold and co: TM Rn an orthonormal parallelization. Iffor some p (1 () , then 2A < 100n8K 2. Example. Let M F r\G be a nilmanifold. Then there exist a sequence of Riemannian metrics { g, } and a corresponding sequence {c w } of orthonormal Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

We prove that a simply connected, complete Riemannian manifold M is rotationally symmetric at p if and only if the exponential image of every linear subspace of Mp is a smooth, closed, totally geodesic submanifold of M. This result is in essence Schur's theorem at one point p, as it becomes apparent in the proof. A Riemannian manifold M is called rotationally symmetric at a point p of M if the isotropy subgroup atp of the isometry group of M is G(«). Recently the rotationally symmetric manifolds received much attention as model spaces for comparison in Riemannian geometry because of the simplicity of the geometric structure. This point was well recognized and systematically developed by Greene and Wu (2). The rotational symmetry condition is a fairly strong one. Therefore there are many clear-cut geometric consequences. The purpose of this paper is to discuss some of these properties, and to show that they in turn characterize the rotational symmetry. Throughout this paper, M is assumed to be a connected, simply con- nected, complete Riemannian manifold of dimension n > 3. M is topologically very simple: M is diffeomorphic to R" or S". One of the interesting geometric properties of M is the existence of closed, totally geodesic submanifolds. In fact the exponential image of any linear subspace of Mp is a smooth, closed, totally geodesic submanifold. In Theorem 3, which is our main result, we shall prove that the converse is also true. Theorem 3 can be viewed heuristically as follows: A totally geodesic submanifold may be envisioned as a fixed point set of some isometry in a certain infinitesimal sense, even though there may be no such isometry. Thus the assumption that the exponential image of any hnear hyperplane is totally geodesic suggests that all reflections with respect to those image hypersurfaces look like isometries. Since the group 0(«) is generated by reflections, it is quite natural to suspect the validity of Theorem 3. Theorem 3 is a version of Schur's theorem at one point, as it becomes visibly clear in the proof. Schur's theorem (e.g. see (3, p. 202)) says that if the sectional curvature of a connected Riemannian manifold N of dimension > 3 depends only on the points of N, then TV is a space of constant curvature. The idea of the proof of Schur's theorem is to show that Xk = 0 for any vector X, where k is the curvature function

The paper contains a vanishing theorem for P harmonic forms on complete rotationally symmetric Riemannian manifolds. This theorem requires no assumptions on curvature. This paper gives necessary and sufficient conditions for existence of L2 harmonic forms on a special class of Riemannian manifolds. Manifolds of this class were called models by Greene and Wu and played a crucial part in the study of function theory on open manifolds [GW]. Throughout the paper M will denote a model of dimension n, i.e. a C X Riemannian manifold such that: (1) there exists a point o EE M for which the exponential mapping is a diffeomorphism of ToM onto M; (2) every linear isometry p: T,oM > To,M is realized as the differential of an isometry 1: M -M, i.e., ?(o) = o and 0.(o) = p. Clearly, M is complete and can be identified with To,M via exp0. In terms of geodesic polar coordinates (r, 9) E (0, oo) x S'~1 M\{o} the Riemannian metric dj2 of M can be written as S2= dr2 + f(r)2 dO2, (3) where dO2 denotes the standard metric on S'1 and the function f(r) is C on [0, oo) and satisfies f(0) = 0, f'(0) = 1, f(r) > 0 for r > 0 (4) (cf. [S, pp. 179-183]). Complete description of the spaces 9C*(M) of L2 harmonic forms is contained in the following THEOREM. Let M be a model of dimension n > 2. Then (i) 9C(M) = {0} for q # 0, n/2, n, [(0) if f(r) dr = , 0(ii) aV(M) ==d 9M, IR if If(r) l r < X0 Received by the editors July 27, 1978 and, in revised form, December 4, 1978. AMS (MOS) subject classifications (1970). Primary 58G99.

Some spectral properties of spherical mean operators defined on a Riemannian manifold are given. As an application we deduce a statistic property of geodesic chains which is interesting from the view point of geometric probability.

We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class C 0 {C_0} . The special case of a manifold of negative curvature is considered as an illustration.