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Actually, Toeplitz operators are pseudodifferential operators of order 0. The Atiyah-Singer index theorem can be formulated in sufficient generality that the Toeplitz index theorem is a special case, …

What all three definitions have in common is that they each try to capture the first order behavior of a smooth function on $M$. The derivative of a smooth function $f$ along a curve $\gamma$ with $ …

Perhaps this is basic knowledge in Riemannian geometry, but I can't seem to figure out the answer. Here is the precise statement of my question. Let $M$ be a Riemannian manifold, $p$ a point in $M$. …

As MO questions go, this one might be borderline - I'm guessing it could be a homework problem in a suitably advanced differential geometry class. I tried asking on math.stackexchange yesterday and i …

In a preliminary version of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88: Neither is one able to prove (or disprov …

There is no standard / classical definition of Gaussian curvature except for surfaces embedded in $\mathbb{R}^3$. I think the pattern of exposition that the OP is asking about is really just an allus …

Your definition is a sort of hybrid of two standard definitions of connection 1-form. These are: A connection 1-form is a Lie-algebra valued 1-form $\omega$ which satisfies $\omega(A^\#) = A$ for a …

A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he appar …

Your question reminds me of the following result: let $X$ and $Y$ be tangent vectors in $T_p M$ such that $|X \wedge Y| = 1$ and let $\gamma$ be the exponential of a piecewise smooth closed curve in $ …

For any connected and oriented $n$-manifold $M$, the sequence: $$\Omega_c^{n-1}(M) \to \Omega_c^n(M) \to \mathbb{R} \to 0$$ is exact, where the first map is $d$ and the second map is $\int_M$. A deta …

My first remark about this question is a little bit pithy - the standard cohomological index formula works only for even dimensional manifolds while Chern-Weil theory is of course more general. For …

I really strongly recommend chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". In this book and its companion volume "Topology, Geometry, and Gauge Fields: Foundations", Naber …

Hitchin showed (here, I think) that there is an exotic sphere which admits no metric of positive scalar curvature. This manifold certainly has the sphere as its topological universal cover, but its s …

Here is an updated formulation of the question, which is more precise and I think completely correct: Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of th …

My question is motivated by the following well-known computation. Let $M$ be an even dimensional Riemannian spin manifold and let $D$ be the spinor Dirac operator on $M$. Lichnerowicz showed that $D …