If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm …

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ suc …

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel tra …

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to whi …

Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathb …

My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — c …

Assuming an appropriate definition of 'curvature', is there a theorem that says: "At least n+1 dimensions are necessary for a particular curvature to exist in an n-dimensional spac …

A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B …

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\i …

I'm a novice in the surgery theory, and I'm encountering the word "(4-dimensional) topological surgery conjecture", which I can't not find the definition. Could anyone help me?

On page 3 of the book before exercise 1.2, is written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free …

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, t …

Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask …

Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular ne …

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)