projects

MATH782 Differential Geometry : some possible project topics 1. Connections in principal bundles [10] : Explore how connections and their curvature are defined for principal G−bundles. An example is the frame bundle of a manifold, which is a principal GL(n, R)−bundle. A connection on the frame bundle induces an affine connection on the tangent bundle, and vice versa. If the affine connection preserves a Riemannian metric, what can we say about the connection on the frame bundle? 2. Cartan’s moving frames : The Cartan formulism is an effective way to do calculations in differential geometry using moving frames. Find out what this means and describe some examples. 3. Isometric immersions into RN [3] : Find out about the existence of isometric immersions of Riemannian manifolds into flat Euclidean space. Cartan-Janet showed the existence for real analytic manifolds in the 1920s. Nash showed existence for smooth manifolds in the 1950s. One can then ask how large the dimension N of the ambient space must be; there are some surprising results for C 1 −manifolds. 4. Isometric immersion as a hypersurface [12] : What can we say about Riemannian manifolds embedded as hypersurfaces? In general this will place strong restrictions on the manifold. For example, CP2 cannot be isometrically embedded as a hypersurface in R5 , even locally. 5. Riemannian submersions [6] : In addition to studying isometric immersions M ,→ M , one can study Riemannian submersions M → M . In this case M is a fibration over M , possibly with some singular fibres, and there is a relation between the metrics g and g. O’Neill’s formula gives a relation between the curvatures of M and M , which in some sense is an analogue of Gauss’s formula for immersions. Some examples to consider are the Hopf fibrations S 3 → S 2 = CP1 , S 7 → S 4 = HP1 , and the S 1 −bundles S 2n+1 → CPn . 6. Einstein manifolds [4] : Manifolds with constant Ricci curvature (meaning that the Ricci tensor is a multiple λ of the metric g) are known as Einstein manifolds. This is a broad topic with a vast amount of literature. Some basic results include a) the multiple λ is constant (a priori it is a scalar function on the manifold) if the dimension is at least three, b) the sectional curvature of an Einstein metric is in fact constant in dimension three. 7. K¨ ahler manifolds [11] : On a complex manifold (i.e., a manifold given by patching together open sets in Cn using holomorphic maps) one can introduce a Hermitian metric. In this context, a K¨ahler metric is a Hermitian metric which satisfies a certain condition known as the K¨ahler condition. Investigate the geometric interpretation of this condition; it is a kind of compatibility between the complex structure and the metric. 8. Ricci-flat metrics [4] : Metrics with vanishing Ricci curvature are solutions of the Einstein vacuum equations, and have interested physicists for decades. Yau’s Theorem (known as the Calabi Conjecture before it was proved) gives a criterion for the existence of a Ricci-flat metric on a compact K¨ ahler manifold. The resulting Calabi-Yau manifolds arise frequently in algebraic geometry. 9. Constant scalar curvature metrics : Perhaps surprisingly, metrics with constant scalar curvature are not rare. Schoen proved that any Riemannian metric on a closed manifold can be conformally rescaled (i.e., we just multiply g by a function f to get a new metric f g) to give a metric with constant scalar curvature. A manifold can also admit a metric with constant 1

positive scalar curvature and another metric with constant negative scalar curvature. Finding K¨ahler metrics on complex manifolds with constant (or zero) scalar curvature is more difficult, and an active area of current research. 10. Isoperimetric inequalities [5, 8]: As is well known, for a fixed perimeter L, the circle is the simple closed curve which encloses the largest area. Since L = 2πr, the circle’s area will be L2 . 4π There are similar formula in higher dimensions: the volume V enclosed by a hypersurface in Rn+1 of “area” A will satisfy n+1 V ≤ cn A n A(L) = πr2 =

where the constant cn depends only on n, with equality for the sphere. In fact, these isoperimetric inequalities also exist in non-flat spaces. Investigate the inequalities that arise in constant positive and constant negative curvature spaces. 11. Bochner methods [9, 12] : The Bochner method can be used to show non-existence of harmonic forms and/or Killing fields on manifolds with certain kinds of curvature. For example, Bochner proved that on a compact oriented Riemannian manifold M with non-negative Ricci curvature, every harmonic 1-form must be parallel. Recall that the first Betti number b1 (M ) can be defined as the dimension of the space of harmonic 1-forms, H1 (M ). Therefore b1 (M ) is bounded above by dimM (an n-dimensional manifold cannot admit more than n independent parallel 1-forms). If the Ricci curvature is positive, even at one point, then all harmonic 1-forms must vanish and b1 (N ) = 0. 12. The Sphere Theorem [6] : As we will see, a compact simply-connected Riemannian manifold with constant positive curvature must be a sphere. More generally, suppose the curvature is positive but not necessarily constant. By rescaling the metric we can assume that the sectional curvature is strictly bounded below by δ and bounded above by 1. We call such a manifold δpinched . The sphere theorem states that a 41 -pinched manifold (i.e., satisfying 14 < K ≤ 1) must still be homeomorphic to a sphere. The result already fails for the weaker hypothesis 41 ≤ K ≤ 1; CPn and various other symmetric spaces provide counter-examples. Also interesting is that the conclusion is only that the manifold is homeomorphic to a sphere, not diffeomorphic. In fact, it is still an open problem whether exotic spheres can admit metrics with positive curvature. 13. Zoll surfaces [3] : A generic geodesic ray on a generic Riemannian manifold will never return to its starting point. Of course, on a sphere all geodesic rays return to their starting points: the geodesics are great circles, and they are all closed. Surprisingly, there are surfaces not isometric to the sphere that also have the property that all their geodesics are closed; find out how these “Zoll surfaces” are constructed. 14. Spectrums [3] : One can define a Laplacian ∆g on functions on a Riemannian manifold (M, g); the (infinite) set of its eigenvalues, {λ1 < λ2 < . . .}, is called the spectrum of (M, g). We also have the length spectrum of (M, g), the (infinite) set of lengths of closed geodesic. Note that if ` is the length of a closed geodesic γ, then 2` is also the length of a closed geodesic, namely the geodesic that traces γ twice. Investigate the relation between these two spectra, for instance, for flat tori. 15. Isospectral manifolds [3] : Two non-isometric Riemannian manifolds are said to be isospectral if they have the same spectrum. Describe examples of isospectral manifolds. In which dimensions can they occur? 2

16. Mostow rigidity [1] : Surfaces of constant negative curvature come in “moduli”, i.e., there exists a (6g − 6)-dimensional family of non-isometric constant curvature Riemannian metric on a surface of genus g. This is no longer true in dimension ≥ 3: constant negative curvature metrics are rigid. Interesting questions remain about the possible volumes of these manifolds. 17. Toponogov comparison theorem [3] : Suppose that a Riemannian manifold has sectional curvature bounded below and above, δ ≤ K ≤ ∆. Now take a triangle with geodesic edges of lengths a, b, and c in M and in the space forms of constant curvature δ and ∆. Toponogov proved a fundamental result in Riemannian geometry that compares the angles in these triangles. 18. Systolic inequalities [3] : Let (T 2 , g) be the two-dimensional torus with some metric g. The systole of (T 2 , g) is the length of the shortest noncontractible curve; denote it by Sys(g). Loewner proved a lower bound for the area of T 2 in terms of the systole, √ 3 2 Areag (T ) ≥ Sys(g)2 . 2 Equality is achieved only when g is the flat metric on the equilateral torus, i.e., the torus coming from the tessellation of the plane by equilateral triangles. For which other surfaces are systolic inequalities known? Are there any systolic inequalities known in higher dimensions, e.g., for three-dimensional tori?

References [1] Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer, 2003. [2] Marcel Berger, Riemannian geometry during the second half of the twentieth century, University Lecture Series, vol. 17, AMS, 2000. [3] Marcel Berger, A panoramic view of Riemannian geometry, Springer-Verlag, Berlin Heidelberg, 2003. [4] Arthur Besse, Einstein manifolds, Spinger-Verlag, New York, 1987. [5] Isaac Chavel, Riemannian geometry : a modern introduction, Cambridge University Press, 1995. [6] Manfredo Perdig˜ ao do Carmo, Riemannian geometry, Birkh¨auser, 1992. [7] Manfredo Perdig˜ ao do Carmo, Differential geometry of curves and surfaces, Prentice Hall, 1976. [8] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 3rd edition, Universitext, Springer, 2004. [9] J¨ urgen Jost, Riemannian geometry and geometric analysis, 2nd edition, Springer, 1998. [10] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry Volume I , Wiley, 1963. [11] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry Volume II , Wiley, 1969. 3

[12] Peter Petersen, Riemannian geometry, Springer Graduate Texts in Mathematics, Springer, 1998.

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