A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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Covariant Derivative Along Arbitrary Vector

The inspiration for this question is section 1.3 of "A Course in Minimal Surfaces," by Colding and Minicozzi. This section has to do with deriving the first variation formula. We are dealing with an $...
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Laplacian on the sphere in polar coordinates

Consider the sphere $S^n \subset \mathbb{R}^{n + 1}$. Let every point on the sphere (except the north pole $(0, 0,..,0, 1)$) get polar coordinates $(r, \theta)$ from the usual stereographic projection ...
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Reference request: Complex structure of Riemann sphere

I'm looking for some resources to find out more about these topics: $S^2$ as complex manifold through the stereographic projection; Relation between $S^1$ and $\mathbb{P}^1(\mathbb{C})$ Homorgaphy ...
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a very basic geometry question about the length of the curve

"the length of the curve is the sum of all its tangent vector" This is what I heard but not able to find in any books. Pictorially, what does this mean? Formally, what would be a proof for this?
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The proof of a Riemannian metric as a $(0,2)$ tensor

I was told that the Riemannian metric is a $(0,2)$ tensor. I have trouble understand this. I know very little geometry, I learnt that the Weingarten map of a hypersurface is a linear map from $T_x\...
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Jacobian of a map between two spaces of different dimension.

I have been asking quite a few questions lately on this same topic, but I am really trying to get a grasp of a lot of the content of my Riemannian Geometry course, so I appreciate the help! on p. 14 ...
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Does bounded scalar curvature imply bounded Ricci curvature?

Does bounded scalar curvature imply bounded Ricci curvature? It is trivial to show the converse, but I do not know whether the above is true. Inspired by a vaguely similar question, I am thinking ...
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Weird notation in do Carmo's Riemannian Geometry

In do Carmo's Riemannian geometry, in ch. 0, sec. 4, ex. 4.2, he discusses the generalization of a 2-surface in $\mathbb{R}^3$ to a $k$-surface in $\mathbb{R}^n$, $k\leq n$ by defining a subset $M^k\...
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How to prove that a manifold with Self-Dual Riemann tensor is Ricci-flat

By self-dual I mean that \begin{equation} *\mathcal{R}_{ab}=\mathcal{R}_{ab}\,, \end{equation} where $\mathcal{R}$ is the curvature 2-form, related to the Riemann tensor $R$ by $\mathcal{R}_{ab}= (1/2)...
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Ricci flow on a compact surface of constant negative curvature

Let us consider a compact surface of constant negative curvature $-1$ and apply the Ricci flow on it. Will the resulting surfaces for short time also have constant negative curvature? If yes, will the ...
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Does every manifold M always admit a Riemannian metric?

In the book "Geometry and Topology for Physicists" by Nash and Sen, in Section 7.6, after showing that the structure group $GL(n,\mathbb{R})$ of a frame bundle $F(M)$ (for a general manifold $M$ of ...
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Change of Hamiltonian with diffeomorphism and in particular, Hamiltonian for the sphere

Let $f:U\to V$ be a diffeomorphism between an open subset $U\subset \mathbb{R}^2$ and $V\subset \mathbb{R}^3$ (not open, of course). Let's put the metric $g=\sum_{i,j}g_{ij}dx_i\otimes dx_j$ on $\...
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Is it true that if $f(p)=p$ and d$f_p=\operatorname{id}$ then $f(\exp_p(v))=\exp_p(v)\ \forall v\in T_pM$?

$M$ is a Riemannian manifold, $f$ a (smooth) function from $M$ to itself, $v$ only vectors for which the exponential map is defined. Then, Is it true that if $f(p)=p$ and $\mathrm{d}f_p=\...
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Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider ...
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96 views

Intrinsic Riemannian distance vs. Euclidean distance

The intrinsic metric $ d_i(x,y) $ on a Riemannian manifold is the infimum over all curves joining $ x $ and $ y $, of the arc length of these curves. For a surface $ S $ embedded in $ \mathbb{R}^3 $, ...
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Continuously differentiable functions in the Heisenberg group

I need some help with the following. We are in the Heisenberg group $\mathbb H^n$: we denote points $P\in \mathbb H^n$ as $P=(x_1,\dots,x_n,y_1,\dots,y_n,t)$ and let $X_j=\frac{\partial}{\partial x_j}+...
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Advanced introduction to riemmanian geometry

After some time of studying differential geometry/topology while avoiding riemmanian geometry I find mysef at a wierd position. I'd like to read a book that contains an introduction to riemannian ...
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24 views

Metric on Heisenberg group arising from CR isomorphism to punctured odd-dimensional sphere

It is well known that odd-dimensional spheres $S^{2n+1} \subset \mathbb{C}^{n+1}$ are CR manifolds, and on removing a point we get a CR isomorphism to the (underlying manifold of the) Heisenberg group ...
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Clarification about the definiton of a metric on a vector bundle

Let $E$ be a vector bundle over a manifold $M$. By definition, a metric on $E$ is a function $g:E \times _M E \to M \times \mathbb{R}$. In wikipedia, they say $g$ is a bundle map. However, it is not ...
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The arc length in Riemannian geometry is well defined (independent from the choice of the coordinates)

Let be $(M,g)$ a connected Riemannian manifold and $p,q \in M$. If $ \phi : [a,b] \rightarrow M$ is $C^\infty$ we define the arc-length of the curve $\phi$ as the quantity: $$J(\phi )= \int_a^b f(\...
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Isomorphisms of bundles

Studying on vector bundles came across the following problem. Question: Let $M_1$ and $M_2$ Riemannian manifolds with $TM_1^{\perp}$ and $TM_2^{\perp}$ their normal of ranks $k_{1}$ and $k_{2}$, ...
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Metric compatibility of dual connection

Let $(M,g)$ be a Riemannian manifold with Levi Civita connection $\nabla$. Then $\nabla$ satisfies a compatibility condition: $(\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y))$ where $(\cdot,-)$ is a Hermitian ...
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Laplace-Beltrami operator as sum of orthogonal projections

Let $M$ be a submanifold of $\mathbb R^l$ with the induced metric. Let $(\xi_\alpha)$ be the standard orthonormal basis on $\mathbb R^l$. For each $x \in M$, let $P_\alpha(x)$ the projection of $\xi_\...
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33 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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Uniqueness of covariant derivative in Do Carmo

2.2 Proposition: Let $M$ be a differentiable manifold with an affine connection $\nabla$. There exists a unique correspondence which associates to a vector field $V$ along the differentiable curve $c:...
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90 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
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53 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And $$-\operatorname{...
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Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
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Computation of Liebracket for Vectorfields assosiated with a Variation of Geodesics

Let $(M,g)$ be a Riemannian manifold, $V \subset \mathbb{R}^2$ be an open subset and $\alpha: V \rightarrow M; (s,t) \mapsto \alpha(s,t)$ a smooth map. for $(s,t) \in V$ one can define $$ \frac{\...
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100 views

Volume density on a Riemannian manifold as a measure

I am having some trouble in seeing exactly how the Riemannian density form gives rise to a measure on $\text{Borel(M)}$. Let $(M,g)$ be a Riemannian manifold. We have the Riemannian density $\mu_g$. ...
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points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
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Diffeomorphism and Orientable double cover

Suppose that the orientable double cover of two homeomorphic surfaces are diffeomorphic, is it true that these surfaces are diffeomorphic?
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37 views

One-sided surfaces and the second variation area formula.

I know how to find the second variation area formula for a two-sided minimal embedded surface in a 3-manifold and the condition for such a surface to be stable. But, what about one-sided surfaces? ...
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Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
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Geodesics of Sasaki metric

I would like to ask the community for a reference on the following question: Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that ...
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Geodesics without a metric

By definition, a geodesic is a mapping $\gamma: I = (0, 1) \rightarrow M$ such that $\nabla_{\dot{\gamma}(t)} \dot{\gamma}(t) = 0$. Here we only need the connection. So, we do not need a metric to ...
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Exponential map on a sphere in spherical coordinates

Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} {...
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Reference on manifolds with boundary

I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please.
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Can we bypass connection?

I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. Can we bypass this ugly object? Only intrinsic quantities ...
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What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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Advanced Differential Geometry Textbook

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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Is such kind of manifold Riemannian? Deforming the metric on the unit square by a weight applied in one direction

If the metric is defined on a bounded subset of the x-y plane,let's say a closed square area $0\le x,y\le1 $, the metric is defined as $$\langle u,v\rangle =\langle (u_x,u_y),(v_x,v_y)\rangle =\langle ...
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Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
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Parallel transform of a vector by Lie derivative

I am new to differential geometry and I learn by myself. It seems that we need something extra called a connection to parallel transport a vector along a curve. But, suppose we have a vector field $...
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About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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163 views

flat manifold, curvature and the circle

A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance ...
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Surfaces obtained by $\gamma$-reduction

$\mathcal{C}$ will denote the collection of all connected compact (not necessarily orientable) smooth 2-dimensional surfaces-without-boundary embedded in $M$ ( here $M$ is a complete Riemannian 3-...
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The Riemann Sphere Interpretation

Is the Riemann sphere anything more than a simple visual tool to help students understand the complex planes, or the behavior of complex valued functions at infinity, limit points etc? Or is there a ...
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Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...