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Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i. …

Consider the Lagrange multiplier $$F(x, t, c) = f(x) + t( g(x) - c).$$ If $g$ satisfies your condition and $f$ is generic, then one can show that $$S:= \{ (x, t, c)\ |\ \nabla f(x) + t \nabla g(x) = 0 …

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of conforma …

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and …

Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology. My question: In what s …

You need a non-Kahler complex manifold. Then the Chern connection will have nontrivial torsion. And the torsion corresponds to the non-closed Kahler form of the metric.