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Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

4 votes
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Geodesic flows and Killing fields

If we have a Killing tensor field $K$ of type $(0,d)$, the function $$I:SM\to \mathbb{R}, \ I(v)= K(v,\dots, v) \ \ \ \ \ \ (\ast )$$ is constant along geodesic flow. This is a well-known knowled …
Vladimir S  Matveev's user avatar
7 votes
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Conformal maps between two given domains

Any conformal map in dimensions $\ge 3$ is necessary a superposition of inversions and isometries (see e.g. the link suggested by Daniele Tampieri in his comment), so it takes the boundary of $D_1$ to …
Vladimir S  Matveev's user avatar
4 votes

If there exists a function on a Riemannian manifold such that its Hessian matrix is the iden...

If a manifold is complete, the existence of the function $\phi$ such that $\nabla_i \nabla_j\phi = g_{ij}$ implies that the metric is flat and that in a `flat' coordinate system such that the metric …
Vladimir S  Matveev's user avatar
3 votes

Curvature of singular Riemannian metric

Under stronger regularity assumptions, an analog of the curvature exists in the weak sense, i.e., in the sense of generalized functions. The stronger regularity assumption is that the metric (in your …
Vladimir S  Matveev's user avatar
7 votes
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Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one …
Vladimir S  Matveev's user avatar
4 votes
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Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$

On the torus $T^2$ with the coordinates $x,y$ and the flat metric $g= dx^2 + dy^2$ take any function $f(x)$. Its hessian is given, after raising the index, by the (1,1)-tensor $f''(x) dx\otimes \fra …
Vladimir S  Matveev's user avatar
6 votes

Which surfaces admit unbounded-length simple geodesics?

Elipsoid does not posess unbounded geodesics with no self-intersection. I do not know a conceptual explanation. My explanation is that (due to integrability of the geodesic flow of ellipsoid) we …
Vladimir S  Matveev's user avatar
3 votes

List of generic properties of Riemannian metrics

Generic metric does not admit a lot of properties some special metrics admit. A good demonstration of this is the examples listed in the question (no multiple eigenvalues) or in the answers of Matheus …
10 votes
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Riemannian metrics preserved by diffeomorphisms

The answer depends on the diffeomorphism. Let me give two examples, both on the standard torus $\mathbb{R}^2/_{\mathbb{Z}^2}$ with coordinates $x,y$. (Example 1:) $$\phi(x,y)= (x+ 1/2,y).$$ Fo …
Vladimir S  Matveev's user avatar
8 votes

Vector Fields in a Riemannian Manifold

I give a geometric explanation of the calculations of Willie, which simultaneously elaborates the suggestion of Deane. The flow of a vector field commuting with Laplacian preserves the Laplacian an …
Vladimir S  Matveev's user avatar
8 votes
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What does it mean that the Hessian is proportional to the metric?

It is known (say, Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans.Amer.Math.Soc. 117(1965) 251– 275; I am not sure that Tashiro is the first who proved it and there were many …
Vladimir S  Matveev's user avatar
18 votes

Is there a global obstruction for a diffeomorphism to be an isometry?

The answer is ``no'', the pointwise condition is not enough. The example exists in dimension 1 already and can be generalized and made arbitrary weird for all dimensions. Consider a smooth functi …
Vladimir S  Matveev's user avatar
1 vote
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Symplectic reduction: from indefinite signature to Riemannian signature

Just take $R^4=C^2$ with complex coordinates $z_1=x_1+ i y_1, z_2= x_2 + i y_2$ and the flat metric $-dz_1 d\bar z_1 + dz_2 d \bar z_2$. It has signature (2,2). As the group of isometries take the …
Vladimir S  Matveev's user avatar
11 votes
Accepted

Perimeter of ellipse: Combination of two geometries

No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close) points is $\lambda …
Vladimir S  Matveev's user avatar
23 votes

Manifolds admitting flat connections

I did not understand the first question Question 1 Are there manifolds with the property that each connection on is never flat? Because one of course can construct, on any manifold, a connectio …
Vladimir S  Matveev's user avatar

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