Seminar on Geometry

Schedule of talks

Abstracts/Notlar

C1: Connection and curvature forms. Mean curvature from an application of algebraic Cartan Lemma.

C2: Higher osculating spaces and higher fundamental forms of a submanifold.

C3: Higher fundamental forms of a submanifold.

C4: Minimal Surfaces.

C5: Elliptic differential systems.

C6: Minimal immersions of the 2-sphere.

C7: The case of N-sphere as the ambient space.

C8: Elliptic differential systems 2.

C9: Minimal immersions of the 2-sphere.

C10: Frenet-Boruvka formulas for the minimal immersion of a 2-sphere.

C11: Frenet-Boruvka formulas of a complex curve in a constant curvature space.

C12: The case of N-sphere as the ambient space.

C13: Minimal Spheres inside Round Spheres.

C14: Minimal Spheres inside Round Spheres 2.

1. Chern, Shiing Shen. - On the minimal immersions of the two-sphere in a space of constant curvature.
   Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., N.J., 1969),
   pp. 27–40. Princeton Univ. Press, 1970.


2. Calabi, Eugenio. - Minimal immersions of surfaces in Euclidean spheres.
   J. Differential Geometry 1 (1967), 111–125.

In this learning seminar series we will make an introduction to the holomorphic curves in the 6-dimensional sphere. Recall that the 6-sphere is the only sphere in higher dimesions which has an almost complex structure. See the Reference-2 for the history of the Hopf problem.

This allows us to talk about holomorphic curves inside it. Since they are complex curves of a complex dimension 3-space, they have an analogue of Frenet-Serre frame equations called Frenet-Boruvka formulas. In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces.

Ingredients of the individual seminars are as follows:

S-1: Hopf Problem.

S-2: Clifford Torus, Willmore and Lawson Conjectures

S-3: SU(3) structure on the 6-sphere (i.e. almost Hermitian).

S-4:Index and nullity of superminimal curves in the 4-sphere.

S-5:Superminimal curves in the 2n-sphere.

S-6: Jacobi Operator.

S-7: Minimal surfaces in the 6-sphere. First and second normal bundles.

S-8: Moving frames in the 6-sphere.

S-9: Moving frames in the 6-sphere 2.

S-10: Holomorphic curvature and torsion.

S-11: Holomorphic Frenet equations.

S-12: Holomorphic interpretation of the Torsion.

S-13: Characteristic classes of bundles in the presence of null-torsion.

1. Madnick, Jesse. - The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.
   Available at the arXiv:2101.09580 (Jan 2021) 34 pages.


2. Agricola, I; Bazzoni, G; Goertsches, O; Konstantis, P; Rollenske, S. - On the history of the Hopf problem.
   Differential Geom. Appl. 57 (2018), 1–9. Here is the Volume and the Event web site.

In this learning seminar series we will make an introduction to the holomorphic curves in the 6-dimensional sphere. Recall that the 6-sphere is the only sphere in higher dimesions which has an almost complex structure. See the Reference-2 for the history of the Hopf problem.

This allows us to talk about holomorphic curves inside it. Since they are complex curves of a complex dimension 3-space, they have an analogue of Frenet-Serre frame equations called Frenet-Boruvka formulas. In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces.

Ingredients of the individual seminars are as follows:

S-1: Introduction. Riemannian vector bundles.

S-2: Laplace operator.

S-3: Geometry of immersed submanifolds.

S-4: Curvature of the normal bundle.

S-5: Mean curvature and minimal varieties.

S-6: The case of Kähler manifolds.

S-7: The case of Kähler manifolds 2.

S-8: The case of Kähler manifolds 3.

S-9: An extension of the Synge Lemma.

S-10: Laplacian of the Shape Operator.

S-11: Mean curvature and minimal varieties.

S-12: Curvature of the normal bundle.

S-13: Projection of parallel vector fields as the Kernel of the Jacobi operator.

1. James Simons. - Minimal varieties in riemannian manifolds.
   Ann. of Math. (2), 88:62–105, 1968.

Kalafat : In this talk, we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. As an application, we deduce the existence of certain special 3 and 4-dimensional submanifolds of G_2 holonomy Riemannian manifolds with special properties. These are called Harvey-Lawson(HL) pairs. Which appeared first in the work of Akbulut & Salur about G_2 dualities. Another application is to the free embeddings. We show that if there is a coassociative-free embedding of a 4-manifold into the Euclidean 7-space then the signature vanishes along with the Euler characteristic. As a more recent application, we exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also compute various cohomology rings. (Joint work with S.Akbulut et al.)

Kalafat2 : Page manifold is the underlying differentiable manifold of the complex surface, obtained out of the process of blowing up the complex projective plane, only once. This space is decorated with a natural Einstein metric, first studied by D.Page in 1978. It is a cohomogeneity-1 manifold. In this talk, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold. Related paper may be accessed from the following:

Kalafat, Mustafa; Sarı, Ramazan. On special submanifolds of the Page space.
Differential Geom. Appl. 74 (2021), 101708, 13 pp.

This page is maintained by  Mustafa Kalafat
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Activities are supported by University of Bonn and Boğaziçi University