"A closed (periodic) plane curve is said to be simple if it is a one-to-one map up to the period (this for a parameterized curve). For a geometric curve, simple means a curve which has the global topology of a circle (in the jargon, it is a differentiable embedding in the plane of the circle seen as an abstract one dimensional manifold). We also will assume the speed never vanishes, indeed that it has unit speed." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Be careful about infinity: we want an infinite number of geometrically distinct geodesics. As a kinematic motion, running twice along a periodic geodesic is different from running only once, but it is not geometrically distinct. For example when working on counting functions one should be careful to distinguish between the counting function for geometric periodic geodesics and that for parameterized ones. The question is difficult because the standard ways to prove existence of periodic geodesics consider them as motions." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Consider geodesics of a surface not as curves on the surface, but as flow lines of a vector field on the unit tangent bundle. To appreciate that such a viewpoint is possible, we can simply write out the geodesic equation, and think of it this way. Now we may ask what portion of the unit tangent bundle is taken up by periodic geodesics. For example, for geodesics of the two dimensional torus, the unit tangent bundle is a three dimensional torus, and the geodesic flow consists of straight lines. Those which are periodic are in fact dense. On the sphere, all geodesics are periodic, so again the periodic geodesics are dense in the unit tangent bundle." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Do not think that everything is known today about plane curves." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"In general, when looking for the minimum of some quantity, the natural approach since the appearance of calculus is to look for minima among the cases where the first variation is zero. Then one studies the case at hand directly or computes the second variation, etc. We did that amply for geodesics and for the isoperimetric inequality. But systoles are not accessible to calculus, in some sense because of their nature, or perhaps because we lack the required tools." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Riemann forged two simultaneous innovations: first, he defined (not too rigorously) a differentiable manifold to be a set of any dimension n, where one can perform differential calculus, change coordinates, etc. In particular, one has differentiable curves, tangent vectors (velocities) of those curves, and a tangent space at each point (i.e. all possible velocities of any curves through that point). Then he asked that a geometry on a manifold be simply an arbitrary positive definite quadratic form on each of those tangent spaces, thought of as the analogue of Gauß’s first fundamental form. One could use the same expression to define length of curves, look for shortest curves, etc." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Thanks to the uniqueness of solutions of ordinary differential equations, in particular for the geodesic equations, periodicity of a geodesic occurs precisely when the geodesic is a loop with the same initial and final velocity. Using the point of view from the unit tangent bundle, and the notion of geodesic flow, the periodic geodesics are precisely the periodic flow lines of the geodesic flow. Note that a periodic geodesic is permitted self-intersections. Those without self-intersections are called simple." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"The inner geometry of curves does not differ from that of straight lines, but the geometry is radically different if we look at the way a curve sits in the plane." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"The natural perspective to take in studying periodic geodesics is to think of a Riemannian manifold as a dynamical system. The viewpoint of dynamical system leads us to many natural questions: (i) What can we say about a geodesic over large intervals of time? (ii) Are nonperiodic geodesics everywhere dense? Or somewhere dense? (iii) How many periodic geodesics are there? (iv) Where are the periodic geodesics situated? (v) How many periodic geodesics have length less than a given number?" (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Today, one of the simplest tools to study the spectrum and the eigenfunctions is the heat equation. It is not the most powerful (the wave equation is more powerful), but the technique is much simpler. Both techniques, incidentally, extend to Riemannian manifolds with little alteration." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"We have implicitly assumed that there are infinitely many periodic geodesics; as we will see below, this is an open question. There is also the notion of geometrically different geodesics: turning more than once along a given periodic geodesic is not considered a different geodesic by a geometer, even if it might be in some sense different for a mechanics expert. The field of geodesic dynamics is dramatically different from that of spectrum geometry. The basic reason is that eigenfunctions are the critical points of the Dirichlet quotient on the (infinite dimensional) vector space of functions on the manifold. Periodic geodesics are the critical points of the length function on the space of all closed curves of the manifold. Sadly enough, this is not a vector space but an infinite dimensional manifold: one cannot play linear algebra with periodic geodesics." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
"Why should a geometer, whose principal concern is in measurements of distance, desire to engage in analysis on a Riemannian manifold? For example, pondering the Laplacian, its eigenvalues and eigenfunctions? Here are some reasons, chosen from among many others. We note also here that the existence of a canonical elliptic differential operator on any Riemannian manifold, one which is moreover easy to define and manipulate, is one of the motivations to consider Riemannian geometry as a basic field of investigation. [...] Riemannian geometry is by its very essence differential, working on manifolds with a differentiable structure. This automatically leads to analysis. It is interesting to note here that, historically, many great contributions to the field of Riemannian geometry came from analysts." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)
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