The Geometry and Topology on Grassmann Manifolds

This paper shows that the Grassmann Manifolds $G_{\bf F}(n,N)$ can all be imbedded in an Euclidean space $M_{\bf F}(N)$ naturally and the imbedding can be realized by the eigenfunctions of Laplacian $\triangle$ on $G_{\bf F}(n,N)$. They are all minimal submanifolds in some spheres of $M_{\bf F}(N)$ respectively. Using these imbeddings, we construct some degenerate Morse functions on Grassmann Manifolds, show that the homology of the complex and quaternion Grassmann Manifolds can be computed easily.Comment: 15 page

Emergent Symmetry on Black Hole Horizons

For a stationary and axisymmetric black hole, there is a natural way to split the fields into a probe sector and a background sector. The equations of motion for the probe sector enjoy a significantly enhanced symmetry on the black hole horizon. The extended symmetry is conformal in four dimensions, while in higher dimensions it is much bigger. This puts conformal symmetry at the bottom of the ladder of symmetries that can arise on black hole horizons in generic dimensions.Comment: 8 page

A Note on Characteristic Classes

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.Comment: 10 page

Fluctuating Black Hole Horizons

In this paper we treat the black hole horizon as a physical boundary to the spacetime and study its dynamics following from the Gibbons-Hawking-York boundary term. Using the Kerr black hole as an example we derive an effective action that describes, in the large wave number limit, a massless Klein-Gordon field living on the average location of the boundary. Complete solutions can be found in the small rotation limit of the black hole. The formulation suggests that the boundary can be treated in the same way as any other matter contributions. In particular, the angular momentum of the boundary matches exactly with that of the black hole, suggesting an interesting possibility that all charges (including the entropy) of the black hole are carried by the boundary. Using this as input, we derive predictions on the Planck scale properties of the boundary.Comment: References added, nature of boundary stress tensor clarified, discussion of statistics refined and a mistake with Hawking temperature corrected, 16 pages; version to appear in journa